A. N. Parshin I. R. Shafarevich (Eds .)

Algebraic Geometry V Fano Varieties

Springer

List of Editors, Authors and Translators Editor-in-Chigf R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow; Institute for Scientific Information (VINITI), ul. Usievicha 20a, 125219 Moscow, Russia, e-mail: [email protected]

Consulting Editor A. N. Parshin, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia, e-mail: [email protected] I. R. Shafarevich, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia, e-mail: [email protected]

Author V. A. Iskovskikh, Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, e-mail: [email protected]

Author und Trunslator Yu. G. Prokhorov, Moscow State University, Department of Mathematics, Algebra Section, Vorobievy Gory, 117234 Moscow, Russia, e-mail: [email protected]

Trunslator S. Tregub, 70115 Tashkent, Uzbekistan

Fano Varieties V A. Iskovskikh

and Yu. G. Prokhorov

Translated from by Yu. G. Prokhorov

the Russian and S. Tregub

Contents Introduct~ion. Chapter

. . .. .

.

..

4

..

..........................................

1. Preliminaries

7

$1.1. Singularitjies ................................................ $1.2. On Numerical Geometry of Cycles ............................ ......................... $1.3. On the blori iMinima Model Program ................ 31.4. Results on Minimal Models in Dimension Three Chapter $2.1. $2.2. $2.3. $2.4. Chapter $3.1. $3.2. s3.3. 53.4. 53.5.

2. Basic Properties

of Fano

Varietits

.......................

23

.............. Definitions, Examples and t,he Sirnplest Properties Some General Res&s ....................................... Exist,ence of Good Divisors in the Fundamental Linear Syst.em. Base Points in the Fundamental Linear Systern ................. 3. Del Pezzo

Va.rieties

7 11 13 17

and Fano Varieties

of Large

...

Index

On Some Preliminary Result,s of Ftr,jita . . .. Del Pezzo Varieties. Drfinition and Preliminary R.esult,s . Nonsingular de1 Pezzo Varieties. St,atement of t,he Main Theorem and Beginning of the Proof. Del Pezzo Varieties with Picard Number p = 1. Continuation of the Proof of the Main Theorem .. . Del Pezzo Varieties wit,11 Picard Nurnl)er p > 2. Conclusion of the Proof of the hIair Theorem . . . . . . . .. . . . ..

23 34 39 47 50 50 53 54 57 62

2

Chapt,er

ColltcIlts

4. Fano Thrccfolds

$4.1. Elementary $4.2. Fa.milies of $4.3. Elementary 34.4. Elementary $4.5. Elementary $4.6. Sorne Other

Rational Lines and R.ational Ra.tional Rational Rational

with Maps: Conies Maps Maps Maps Maps

p = 1

65

..

Preliminary Results on Fano Threefolds . wit,11 Center along a Line with Center along a Conic with Center at a Point . . .

Chapter 5. Fano Varieties of Coindex The Vector Bundle Method

3 with

p = 1: ..

. .

65 71 76 86 95 101

. 104

$5.1. Fano Threefolds of Germs 6 and 8: Gushrl’s Approach $5.2. A R.eview of Mukai’s Results on the Classifica.tion of Fano Manifolds of Coindrx 3 .

108

Chapter 6. Boundedness and Rational Corlrlr:c.tctlnrss of Fano Varieties .................................................

11G

36.1. Uniruledness ............................................... s6.2. R.ational Cormectedness of Fano Variet,ies

116 120

Chapter

7. Fano Variet,ies

......................

wit)11 p > 2 . . .

. 128

57.1. Fano Tin-cefolds with Picard Number p > 2 (Survey of R.csult,s of Mori and h1uka.i .. . $7.2. A Survey of Results about Higllc?r-clirnensional Fano Va.rieties with Picard Number p > 2 Chapter

8. Rationality

58.1. Intermediate $8.2. Intermediate $8.3. The Brauer Chapt,er

Questions

for Fano Varieties

I. ................

..................... .Jacohian and Prym Va.rictics Ja.cobia.n: the Abel-Jacobi Map ................... .................... Group as a Birat,ional Invariant

9. Rationality

Questions

for Fano Varieties

II ................

$9.1. Birational Allt,olrlorl)llislrls of Fano Varieties .................... $9.2. Decomposition of Birational Maps in the Context ofnIoriTllcory.............................................17 Chapter 10. Some General Constructions of R.ationality ............................................... and Unirationality $10.1. $10.2. 310.3.

104

......................... Sonic Constructions of Unirationality ...................... rJrliratiorln1it.y of Complct,c: Int,crscctions .................... Sonie Geucral Constructions of Rationality

128 141 153 153 162 166 170 170 8

183 183 188 191

3

Contents

Problems

.............

On the Classification of Three-dimensional Q-Fan0 ........................................... Generalizations .................................... Some Particular Resuks ...................................... Some Open Problems

Varieties

Chapkr $11.1. $11.2. $11.3. 5 11.4.

11. Some Part,icular

Chapter $12.1. $12.2. $12.3. $12.4. $12.5. $12.6. $12.7. 512.8.

12. Appendix:

Tables

Results

and Open

.. . .

.

.............. Del Pezzo Manifolds. Fano Threefolds with p = 1. ........ Fano Threefolds wit,h p = 2. ........ Fano Threefolds with p = 3. ........ Fano Threefolds with p = 4. ........ Fano Threefolds with p > 5. ........ Fano Fourfolds of Index 2 wit,h p > 2 ............. Toric Fano Threefolds

References

.

.. . .. . ..

196 .....

..

.. .. .. . . ....

. .. . .. . . .

..

196 ,203 208 .2p.214

.. . .. .

214 214 217 220 223 . 224 225 226

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...227

A’

Introduction

Introduction This survey continues the series of surveys devoted to the classification of algebraic variet,ies (Shokurov (1988), Shokurov (1989), Danilov (1988) Danilov (1989)) Iskovskikh-Shafarevich (1989)) Kulikov-Kurchanov (1989)). It tlcals__---with Fano varieties of dimension three and higher. The general classifica.tion problem was stated and partly advanced in the classical research of Italian geometers. In the last two decades the classification theory developed rapidly thanks to the new Tori theory of minimal models. It is based on the remarkable ideas and results due to S. Mori on cxtremal properties of cones of effective orle-dimensional cycles (Tori (1982)) Tori (1988))) using which the concept, of a minimal model playing the central role in the classical birational classification of surfaces (see Iskovskikh-Shafarevich (1989)) was extended to varieties of higher dirnension. Within the framework of the theory arises the category of projective varieties with some admissible singularities: terminal canonica,l, log canonical and others. A minimal model in the sense of Tori is defined to be a normal projective variety with a numerically effective canonical divisor. According to the Tori Minimal Model Program, which is completely carrird out, in dirnensions < 3 and partly in dimensions > 4 (see Mori (1982), Reid (1983a.), (1988). Koll&r et, K;~warnat,a-nIatsuda-M,2tsllki (1987)) Cl emens-KollBr-Tori al. (1992), Wilson (1987a)), every irreducible algebraic variety over an algebraica,lly closed field of cha,racteristic zero is birationally equivalent either to a minimal model (if its Kodaira dimension > 0) or to a fibration over a variety of srnaller dimension (in particular, over a point) with rational singularities with the general fiber being a Fano variety (in this case the Kodaira dimension of the initial variety equals -c~). Therefore the Mori program establishes the important role t,hat Fano varieties play in the birational classification of algebraic varieties. They are defined to be varieties with ample anticanonical class and form a, subclass of varieties of Kodaira dimension --oc. The only one-dimensional Fano varieties arc the projective lint over an algebraically closed field and a conic over an arbit,ra.ry field. Two-dimensional Fano varieties are de1 Pezzo surfaces (see the survey of Iskovskikh-Shafarevich (1989)). In connection with the problems of rationality and unirationality, G. Fano st,udied at t,he beginning of the century the class of variet,ies with canonical curve-sections (see Fa,no (1908), F ano (1915), Fano (1930), Fano (1931). Fano (1936), Fano (1942) j Fano (1947)). Contemporary authors contimlcd this st,udy, t,aking for the definition of the class of varieties the ampleness of the anticanonical sheaf. G. Fano did not restrict himself to considering only nonsingular varictics, but for the present only nonsingular three-dimensional Fano varieties a.re classified in contemporary works. Although the problems of rat,ionality and unirationality st,ill remain important (and very difficult),

Introtluction

5

at present it is the general problems of the structure theory that are of prime interest, namely, the classification of Q-Fan0 varieties with admissible singularities in dimension three and higher, the problem of boundedness of the degree, solved for nonsingular Fano varieties of any dimension in Kollk-Miyaoka-Mori (1992c), Nadel (1991), Campana (1991a), the problems of uniruledness and rational connectedness, also solved for nonsingulaa Fano varieties of any dimension (see Miyaoka-Mori (1986), Kollk-nliyaokaMori (1992c), Campana (1992)), the study of Fano varieties with additional structures (PT-bundles, toric varieties, and others; see Batyrev (1981), Demin (1980), Szurek-WiBniewski (199Oc), Wianiewski (1989b), WiSniewski (1993)). There are several rather complete expositions of the classification theory for nonsingular three-dirnensional Fano varieties (see, for example, Iskovskikh (1979a), Iskovskikh (1988)) Murre (1982)) Mori-Mukai (1986), Mori-Mukai (1983a), Mukai (1992a)). S’m g u 1ar F ano varieties and Fano varieties of higher dimension have been studied in the last decade. The goal of the present survey is to encompass as far as possible these separate results and to highlight the main directions and methods of research. We do not, include in this survey t,he well-known (actually classical) results on two-dimensional Fano varieties, t,ha,t is, nonsingular de1 Pezzo surfaces (see, for example, Naga.ta (1960)) Manin (1972))) and on de1 Pezzo surfaces with ca.nonical (see Du Val (1934): Demazme (1980)) Hidaka-Watanabe (1981)) Brenton (1980)) and log terminal singularit,& (see Alcxeev (1988)) Alexeev-Nikulin (1989), Nikulin (1989a), Nikulin (1988), Nikulin (1989). Alexeev (199413)). We do not touch arithmetic results for Fano varieties (see Manin-Tsfasn1a.n (1986): Batyrev-Marlin (1990), Marlin (1993))) and the few known results in characteristic p > 0 (see Ballico (1989), Serpico (1980), Shepherd-Barron (1997)). The ground field k is assumed to be algebraically closed and of characteristic zero. The Russian version of this survey was finished in 1995 and unfortunately rnany works on this subject appearing later were not included in it. The survey begins with a brief exposition of some points of the Mori theory of minimal models of algebraic varieties, which assumes a central place in contemporary algebraic-geometric research. This is the contents of Chapt,er 1. In Chapter 2 we give the basic definitions a,nd examples, and formulate the simplest properties of Fano varieties which can be irnmediately deduced from the definition and general theorems such as the Riema.nn -Roth theorem, vanishing theorerns etc. We also include in this chapter some general results on equations defining varieties connected with Fano varieties (canonical curves, varieties of minimal degree, intersections of quadrics). At the end of the chapter we reproduce some results on the existence of good divisors in anticanonical linear systerns and on their base locus. Chapter 3 is devoted to the description of the results due to T. Fujita on the classification of polarized dcl Pezzo varieties connected with n-dimensional Fano varieties of index r~ - 1. We give also the proof of the main classification t#lieorem.

6

Introduction

In Chapter 4 we present the classification of tllree-dinlerlsional Fano varieties with Picard number p = 1. The classification is based on t,he generalizat,ion of the classical rnethod due t,o G. Fano of a double projection from a line, a conic, etc. with t,he use of Mori theory. In Chapter 5 we describe the approach of N. P. Gushel and S. Mukai to the classification of Fano varieties with p = 1 using vector bundles. The method of vect,or bundles rnakes it possible t,o obtain a new proof of the classification theorem for three-dimensional Fano varieties with p = 1. This was done in Mukai (1988), Mukai (1989), Mukai (1992a), and in Gushel (1982), Gushel (1983), Gushel (1992) (only for genus g = 6 and 8). Results on the uniruledness, the rational connect,cdness and t,he boundedness of the degree for n-dimensional Fano varieties are presented in chapter 6. Chapter 7 is devoted to the classification of Fano varieties with p > 2. In the first section we describe the Mori-Mukai classification of three-dimensional Fano varieties with p > 2. In the second section we present some results related to the classification of Fano varieties of higher dimension with p > 2. The problems of rationality for Fano varieties are discussed in chapters 810. We discuss briefly the basic methods for proving the non-rationality. In Chapter 8 we consider the method of intermediate Jacobians due to C. Clemens and Ph. Griffiths and the method connected with the Brauer group due to D. Mumford and M. Arttin. In Chapter 9 we consider the method of factorizat,ion of birational maps (the classical NoetherFano method and its generalization in the context of Mori theory). In Chapter 10 we collect the known general constructions of unirationality and rationality for Fano varieties and some concrete results as well. In Chapter 11 we note some generalizations of Fano varieties known to us, describe some separate results not included in the main text, and give a list of open questions and problems. The classification tables for de1 Pezzo varieties and nonsingular threedimensional Fano varieties are placed in Chapter 12. The first author worked on the final version of the survey during his visit to the Universities of Pisa and Genova. He would like to express his deep gratitude to the Departments of Mathematics and especially to Professors F. Bardelli; I. Bauer, F. Catanese and M. Beltrarnetti for their hospitality and the opportunity to work in excellent conditions. He also thanks the Italian Consiglio Nazionale delle Ricerche (CNR) for the financial support. The second author thanks the fund “Pro Mathematics” for the financial support. The present work was also partly financed by Grant No M30000 from the Internat,ional Science Foundation. and by the Russian foundation for fundamema researches (project 93-011-1539). The authors would like to thank N. A. Valueva for her help in preparation the manuscript.

$1.1. Singularities

Chapter 1 Preliminaries fj 1.1. Singularities Let X be an irreducible normal projective variety of dimension dimX = d over an algebraically closed field k of characteristic zero. In this survey we shall use the following notation (see Kawamata-Matsuda-Matsuki (1987). ClemensKoll;ir-Mori (1988)). Z,,.(X) is the ,973~~ ofr-dimensional cycles on X, that is, the free a,belian group generated by closed irreducible subvarieties of dimension T, 0 < T < is the gron2, of Weil divisor3 or1 X. d - 1. In particular, Z,-,(X) Div(X) is th e g roup of Curtier divisors on X. There is a natural irljection (since X is normal) Div(X) - z,i-l(X) , the image consists of those divisors which locally in a neighborhood of every point can be determined by one equation. Elements of the group Z(i-1 (X) @Q are called Q-divisors, and elernent,s of the group Div(X) N Q are called QCurtier divisors. Pit(X) denotes, as usual, the Picard group, that is, the group of classes of Cartier divisors with respect t,o linear equivalence. This group is naturally isomorphic to the group of invert,ible sheaves (or, equivalently, line bundles) on X up t,o isomorphism. > A Q-Cartier divisor D E Div(X) @Q is said to be big if h”(X, Qx(mD)) const ,ld for (sufficiently large) 0 such that, 7n.D t Div(X). effectiw) if A Q-Cartier divisor D E Div(X) @ Q IS . ca.lled nef (numerically D. C > 0 for every complete curve C c X. The intersection number is defined as a rational number in the following way: let m E Z be an integer such that m,D E Div(X); t,hrn 1 D t C := - deg(Ox(rnD)Ic:) m

For any invertible sheaf L E Pit(X), the degree cleg(Llc) is defined to be equal to deg u*L, where V: c + C is the normalization of the curve C. equivalent, which is written Cycles z, z’ c 21 (X) are said t,o be numerically as z G z’, if L z = L . 2’ for every L E Pit(X). By duality, one can define numerical equivalence in Pit(X). The pairing Pit(X) x 21 (X) + Z induces a perfect pairing N’(X) x l&(X) + R ) (1.1.1) where N’(X) For Pic(X)/(mod

:= Pic(X)/(mod E) @R, and Nl(X) := Z,(X)/(mod N;(X). -) R Q, we use the notation

-)

@ IR.

8

Chapter

1. Preliminaries

Let w be a rational differential form of degree d on X. Then one can define the Curtier &7kor of this form (w)lu on the open dense smooth subset U = X - Sing X (codim X > 2 because X is normal). It can be extended to a Weil divisor on the whole X; the class of this Weil divisor is called cunonical arrd is denoted by Kx or simply by K. A variety X is called Gorenstein (or its singularities are called Gorenstein) if it is Cohen-Macaulay, and the dualizing sheaf wx is invertible. In such a case KX E Pit(X), that is, Kx is represented by a Cartier divisor (the converse in general is not true, see I&ii (1987), Ishii (1991)). If mKx E Pit(X) for some integer rn: then X is called Q-Gorenstein. The minimal positive integer 7n with this property is called the (Gorenstein) index of Kx (or X). All srnooth varieties are, of course, Gorenstein. Example 1.1.1. Let F4 C p” be the Veronese surface, that is, the image of p2 in p” under the map determined by the complete linear system of tonics 0~2 (2) i and let X c p” be a cone over Fd. Then Pic( X) = Z. H, where H is a hyperplane section. Denote by E c X the Weil divisor which is the cone over the image of a line 1 E p2 in F4. Then -Kx = 3E @ Pit(X), so the singularity in the vertex is not, Gorenst,ein. But 2Kx = -BE = -2H E Pit(X), which means that X is Q-Gorenstein of index 2. Note that locally near the vertex, X can be represented as c:“/r, where T is the involution (z:, y, Z) 4 (-z, -y, -z). The expression (dz A dy /? dz)” = (dx” A dy2 r\ d~~)‘~/64:~:~y~z~ defines a. 2canonical form on X. If f:X + X is t,he blow-up of the vertex P E X, and if F = f-‘(P) CY I?2 IS the exceptional divisor, then X is norrsingula.r, and Kz = f*h’x + ;F. Definition cd

1.1.2.

(respectively,

and for every resolution Ei c X’, the following

Q-Gorenstein, tional

A normal variety log terminal,

termdinal,

dkisors

mKxJ (m is the index of X) Usually this formsulu

with

X

is said

log

canonical)

conditions

most

cunoni-

f: X’

4

X

if it with,

is

ezcep-

hold:

+ c

aj7 > 0 (respectively, by m, and is

at

singu1uritie.s

of singularities

= .f*(rnKx)

is divided

to have

ai

written

(1.1.2)

a,E, >

0, 2

>

-1,

2 > -1).

as

for canonical (respectively, Q,, > 0 for terminal, CY~ > -1 for log (u, > -1 for log canonical) singularities. The numbers (Y, are culled discrepancies at E,; they depend only on X and the (proper images of) dkuisors E,, that is, they do not depend on the choice of resolution.

where

a,

> 0

terminul,

Let

u.nd

wx

=

O(Kx)

integer

i, we denote

to kill

torsion

be the canonical (dua.lizing) sheaf. For any positive by WY the double dual sheaf of the sheaf LJ$’ (it is taken

and co-torsion,

see Reid

(1980b)).

Then

WY is a torsion-free

Singularities

fil.1.

9

sheaf of rank 1: it is locally free if and only if i = am for some integer where rn! is the index of Kx. If X has at most canonical singularities, f: X’ + X is a resolution, then f*(w$2,) = WY for i 2 0. Proposition

1.1.3

(see, for example,

Clemens-KollBr-Mori

(1988)).

(i) In dimension two, terminal points are nonsingular. (ii) Two-dimensional canonical singularities are exactly the Du are also called rational double points). Locally in the complex can be determined by one of the following equations: A,, :

xy + znfl

D,, :

J? + y2z + iF1

:

E(j

= 0,

a > 0, and

Vu1 ones (they topology

they

n > 1 ; = 0,

n > 4 ;

X2 + lJ3 + Z4 = 0 ;

E7 :

x2 + y3 + yz’

E8 :

x2 + y” + z5 = 0

(1.1.3)

= 0 ;

Minimal resolutions of these singularities are described by the corresponding Dynkin diagrams; irreducible exceptional curves are represented by vertices, and two vertices are connected by an edge if t,he corresponding curves intersect. The number of vertices of the diagram is equal to n. There is a complete list of terminal singularities in dimension 3 (refer t,o Mori (1985), Ried (1987), Koll& (1991)): Theorem (i) (ii)

(iii)

1.1.4.

Three-dimensional terminal singularities are isolated points. A three-dimensional hypersurface (i. e. Goren,stein) singsularity is terminal if and only if it is isolated and is de$ned by an equation of the form:

where

g is

called

(compound

Every

three-dimensional

persurface where

one

rn

is the

(x1,

(1.1.3).

terminal

index

. . . . xN)

t)

is

usually

Ried (1987)).

a quotient

of

some

hy-

a canonical (m : l)-cover, point) by sorne cyclic gro’up.

at the sin,gulu,r

4

( 0, are true (see Elkik (1981): Kawarnata-MatsudaMat)suki (1987)). L 0 g canonical singularities are not necessarily rational even in dirnension 2 (see Koll6r et a.l. (1992)). (iv) Terminal singularities form the least possible class of singularit,ies involving which the Tori Minimal Model Progra,m is stated and can be true (as in dirnension 3). Canonical singularities are exactly those which arise on canonical models. (v) In the general Minimal Model Program (Kawamata-Matsuda-Matsuki (1987), Koll& et al. (1992)), more general types of singularities are defined and used. Namely, let D = C Q, D, be a Q-divisor on a normal variety X (D; are irreducible Weil divisors) such that Kx + D is a Q-Cartier divisor. Then for every resolution f: X’ + X we have: Kx,

=.f*(Kx+D)+~n,F;,,

where E1 are not necessarily only exceptional A pair (X, D) or Kx + D is called:

divisors.

if (I, > 0; if (Ye > 0;

termid canonical purely

(1.1.4)

aJ EQ,

if 0’1 > -1; if a, > -1.

log terminal

log canonical

A pair (X, D) 1s called Ku*warrmtu log terrninul if (X, D) is purely log terminal, and QL < 1 for all i. The Q-divisor D = C a,D’, is called the boundary/ if 0 < Q, < 1 vJ1;. The pair (X, D) is called a log variety, and Kx + D is called a log canonical dkisor. In terms of discrepancies of only exceptional divisors, formula (1.1.4) can be rewritten in the form Kxf

+ (f-‘)*(D)

= f*(Kx

+ D) + ~cY,F;,,

where t,he star in the subscript denotes thr proper A resolution f: X’ 4 X is called a log rwolution,

oi E Q ,

irnage of D as a Weil divisor. of the log variet,y (X, D) if

51.2. On Numerical

Geomet,ry

of Cycles

irreducible components of t,he support Supp( (f-l)* intersect transversally. Refer to Shokllrov (1992), Kollrir (1997) for different variants of t,he definition this more general situation.

51.2.

On Numerical

Geometry

11

(D)) a.re nonsingular and Koll&r et al. (1992) and of log terminal objects in

of Cycles

Let X he a normal projective variet,y of dirnension d. A cycle z = C n,z, E &(X) is called eflectiue if r~,, > 0 V’i. R.ecall t,hat we denote the numerical equivalence of l-cycles wit,h respect to intersections with Cartier divisors (and, by duality, the numerical equivalence of Q-Cartier divisors) by the symbol E. A variet,y X is called Q-factorid if some integral multiple of every Weil divisor is a Cartier divisor, that is, if 2,1-l(X) @ Q = Div(X) @ Q. The following notation is sta.ndard (see Clemens-Kolldr-Tori (1988), Kawamata-Matsuda-LZatsuki (1987)): N(X) := Z,(X)/(mod E) c-3R; NE(X) is the least convex cone in N(X) containing all effective l-cycles; NE is the closure of NE(X) in tk IC real topology; this is t,he so-called hrlori corle of x; NS(X) is the N&on-Seve~ri grog of classes of Cartier divisors with respect to algebraic equivalence: is the Picard number. p(X) := rk(NS(X)) = c1’IrnN N(X) Note that a mlmerically effective Q-Cartier divisor D is big if and only if D” > 0 (the self-intersection index of a Q-Cartier divisor is defined as a rational number). Assume now that X is Q-Gorenstein. A half-line R = Iw+[z] c NE(X), z E Z,(X) x IIR is called an extremal ruy if: (i) -Kx z > 0, and (ii) from z1 + zz E R, 21: z2 E NE(X) it follows t,hat z1 E R and zz E R; this means that the ray R lies on the boundary of cone NE(X). A rational curve C c X is called an eztremnl curve if Iw+ [C] is an extremal ray and 0 < -Kx .C < df 1. The important invariant of an extrernal ray is the number p(R) = inf{-Kx C 1 C c X is a rational curve whose numerical class is [C] E R}. This number is called the length of the extremal ray R. An extrernal ray R is called numeri,call:y eflective if C D > 0 for every effective irreducible Q-Cart,ier divisor D and a curve C such that [C] E R. For every Q-Cartier divisor D, WCset,

NED(X)

= {z E NE(X)

/ D. z > 0} ;

Chapter

12

1. Preliminaries

in particular, we shall call NEK(X) the positive part of the Mori cone, and NE-x(X) = {Z E NE(X) ( Kx z < 0) the negative part. By definition, all extremal rays lie in the negative part of the Mori cone. The following importantj result was proved by Kleirnan (1962). Theorem D on a variety

1.2.1 X

(Kleiman’s is ample

if

crit,erion for ampleness). A @Curtier and only if D. z > 0 for every z t m(X)

For the anticanonical divisor -Kx on a three-dirnensional ampleness criterion takes on the following simpler form.

di~~isor

- (0).

variety

X,

the

Theorem 1.2.2 (Matsuki (1987)). Let X be a complete normal variety of dimension 3 with at most cano,nical singularities. Assume tho,t / - mKxI # 0 for some integer m > 0 (that is, K(-K~) 2 0, whesre n(D) denotes the Kodnira dimension of a divisor D, see the definition below). Then the divisor -Kx is ample if and only Q -Kx . C > 0 for every irreducible curue C E 21(X). The irnportant notion in the Minirnal Model Program is the numerical sion of a numerically effective Q-Cartier divisor D E Div(X) 18 Q: v(D)

:= max{m

1 D’”

dirnen-

$ O}.

If Kx is nef, then v(Kx) is called the numerical dimension of the variety X and is often denoted by u(X). The numerical dimension is closely related to the Kodaira dimension. Recall the definition of the latter. For any Cartier divisor D E Pit(X), denot,e by qDl: X+pdirn IDI. as usual, the rational map determined by the complete linear systern IDI. The Kodairu D-dimension 6(X, D) is defined as follows (it is also called the Iitakn D-dimension):

K(X,

D)

max{dim

:=

yllrZ~i (X)}

1 -cc

if IrnDI # 0 for some integer otherwise.

m > 0;

We remark that, &(X, D) can be characterized by the property: there exist, cli,p > 0 and mo E Z, mg > 0, such that the following inequalities hold for m >> 0: arnK 5 h”(X, Ox(mm~D)) < ,b’m” Let R(X, the tensor K(X,

D) :=

@ H”(X, &(mD)) be a graded 711)O multiplication of sections; then

D) =

(transcendence 1 -cc

degree

of R(X,

D))

algebra

- 1

with

respect

t,o

if R(X, D) # k; otherwise.

A divisor D E Div(X) is big if 6(X, D) = d = dimX. The Kodak dimension h(X) of X is defined t,o be equal to ~(x’. where X’ is any complete nonsingular variety birationally isomorphic

Kx,). to X

51.3. On the Mori

Minimal

Model

(for example, any desingularization of X). The to be R(X) := R(X’) := @ H”(X’,0x~(mKx,)). 11120 K(X)

:=

(tra.nscendence 1 -32

degree

Prograrn

13

ring of X is defined We have

cunonical

of R(X))

- 1

if n(X) # Ic; otherwise.

The variety X is said to be of general type if K(X) = dirnX, that is, if Kx, is big for some nonsingular model X’ of the variety X. Comparing K(X) with t,he numerical dirnension U(X), we see that always 6(X) 5 y(X), in particular, K(X) = V(X) f or varieties of general type with nef Kx.

$1.3. On the Mori

Minimal

Model

Program

It is well-known for nonsingular projective surfaces that a minirnal surface can be obtained from any other surface by contracting exceptional curves of the first kind ((-l)curves). As a result, the following theorem is true (see: for example, Griffiths-Harris (1978)). Theorem

1.3.1.

Every

minim,al

surfuce

X satisfies

one of the following

conditions:

(i) Kx is nef, that (ii) X is a PI-bundle (iii) X = p2. Theorem u(X) no

=

K(X),

base points

is,

Kx over

. C 2 0 for a smooth

any curve

curve

C c X;

r;

1.3.2. Let X be u smooth projective surface with and the complete linear system IrnKxI is free and fixed components) for n, >> 0.

n,ef

Kx.

(th,at

is,

Then it

has

The new approach to the Minimal Model Problem in higher dimensions was suggested by Mori (1982) and is called the Mori Minimal Model Program for algebraic varieties (see Reid (1983a), Clemens-Kollar-Mori (1988)) KawamataMatsuda-Matsuki (1987)). Th e essential ,. L difference from dimension 2 is that one should admit singularities on minimal rnodels. The rnost difficult part, of the program is the problem of the existence of so-called flips. In dimension 3 this problem was solved by Mori (1988) and in a, more general context by Shokurov (1992); see also Kollar et al. (1992). In dimension d 2 4: it still remains open. An analoguc of Theorern 1.3.2 for higher dirnensions is the abundance conjecture. For dimension 3 this conjecture was proved by Miyaoka (1988), Miyaoka (1988a) and Kawama.ta. (1992c). In this and the next section, we shall give a brief account of results related to the Minirnal Model Program. Definition (i)

1.3.3.

Let (Y E II% be a real > (2, and LNJ denotes

number. the

greatest

The

symbol integer

[trl 5 a,

denotes i. e. LcxJ =

the [a]

least integer is the

usual

Chapter

14 integral

(ii)

part

1. Preliminaries

ofru.

Let D = c u,,D;

For the following

be a

f&ZhJiSO?-.

theorem,

we

see (Kawamata.

set

(1982),

Viehweg

(1982)).

Theorem 1.3.4 (Vanishing Theorem). Let X be a nonsingular proper algeb*ruic variety, and let D = C CI!D, be a nef and big Q-divisor. Ass!ume that th,e support of th,e fractional part {D} I1a.s only normal cro.ssings. Then H’(X,Ox(Kx+[Dj))=O,

V’i>O.

If D is an intjegral ample divisor, this is the well-known Kodaira vanishing theorem. This generalization of the Kodaira. vaniskring theorem is due to Y. Kawamata and E. Viekiweg. It looks rather artificial but is widely used in applications. Theorem 1.3.5 (Non-vanishing Theorem. Shokurov (1985)). Let X be a nonsingular projecti’ve variety, and let D be a nef Cartier disvisor. Let G be a Q-divisor such that [Cl is effective. Assum,e th,at the Q-&visor uD + G ~ h’x is ample for some u E Q, a > 0, and the support of the fructio~nal part {G} has only normal crossin,gs, that is, all its irreducible components are nonsingular clnd intersect transversally. Then H”(X, for

any

su#ki:ently

0x(mD

large integer

+ [Gl))

# 0

m >> 0.

The divisor G is involved here for some tjechnical reasons important applications. It will not simplify the proof if one takes G = 0. By tkir preceding vanishing theorem, H’(X,

C’x(mD

+ [Gl))

= x(X,

Ox(mD

for

+ /Gl)).

For the following theorem, see (Reid (1983h), Shokurov (1985)! (1984), Clemens-Kollar-Mori Matsuda-Matsuki (1987). K awamata

Kawamat’a(1988)).

Theorem 1.3.6 (Base Point Free Theorern). Let X be a projectke variety with at most log terminal sin~gularities. Let D be u n,ef Cartier divisor such th,at the Q-divisor aD - Kx is nef and big for some u E Q, a > 0. Then the &ear system, lnD1 is free for all rn >> 0.

51.3.

For the following Kawamata-Matsuda-Matsuki

Theorem with at most and let

On

the

Mori

Minimal

Model

Program

15

theorem, see (Kawamata. (1984), Kollk (1987)) Clemens-Kollar-Mori (1988)).

1.3.7 (Rationality Theorem). Let X be u projective log terminal singularities. Let H be an ample Cur-tier r := max{t

E Iw 1 H + tKx

Theorem

variety divisor,

is nef}

Assume that Kx is not nef. Then r is a rational n,um,ber which sented as U/U, where 0 < II < (indexX)(dimX + 1). For the following theorem see Kawamata Matsuki (1987)) Clemens-KollBr-Mori (1988).

(1984),

(1984),

cun be repre-

Kawamata-Matsuda-

1.3.8 (Cone Theorem).

(i) Let X be a projective variety with at most log terminal singularities. Then all the extremal rays of the (closed) Mori cone NE(X) form a discrete subset in the open half-space {z E N1 (X) 1 Kx z < O}; (ii) m(X) = NEK(X) + CR,, fluhere {R,} is the set of all extremal rays. This is the main theorem in the Minimal Model Program. One proves it in the following succession: 1.3.5 + 1.3.6 + 1.3.7 + 1.3.8. For the following theorem, see (Clemens-Kollk-Mori (1988), KawamataMatsuda-Matsuki (1987)).

Theorem 1.3.9 (Contraction Theorem). Let X be a Q-fuctoriul projective variety with at most terminal (respectively, canonical, log terminal) singularities. Then the following assertions are true. (i) For every extremal ray R c NE(X), th ere exists a contraction morphism j’: X + Y such that an irreducible curve C is mapped to a point if and only if [C] E R. 0 ne can require that f,c?,y = 0y, and th,en f and Y are uniquely determined. (ii) One has only the following possibilities for f and Y: a) the morphism f is birutional, and its exceptional locus is CLdivisor. In this case Y is again Q-factorial with terminal (respectively, canonicul, log termina,l) singularities; such an f is called a divisorial contraction. b) f is birational, and its exceptionul locus has codimension > 2 *in X. In this case Y is not Q-Gorenstein; such an f is called a small extremal contraction. c) dim Y < dim X; the generic fiber F = f-‘(q), where rl E Y is the generic point, h,as terminal (respectively, cunonicul, log terminal) singularities and an ample anticanonical class - Kx . In such a case f is called a Fano contraction. In all the cases, there

exists

the following

0 4 Pit(Y)

+ Pit(X)

exact --t Z

sequence: (1.3.1)

Chapter

16

1. Preliminaries

The most satisfactory assertion is part a). This is a generalization of contractions of (-l)-curves on surfaces to arbitrary dimensions. In this case the variety Y also satisfies the conditions of the theorem, but rkPic(Y) = rkPic(X) - 1 by forrnula (1.3.1). A ssertion c) reduces the study of X to the study of varieties F and Y of smaller dimension. In particular, it can happen t,hat dirnY = 0, and then X is a so-called Q-Fano variety with terminal (or canonical) singularities. Even in the case of dimension 3, t,he study of such singular Q-Fan0 varieties is far from being completed. The most difficult part (and new in comparison with the case of dimension 2) is assertion c). To treatt this case, nrw birational operations should be involved, so-called flips, which we shall briefly discuss in the next section. Now we come t,o the basic Definition

1.3.10. Then:

singulwities.

Let X he a Q-factorial

uariet:y

with

at

most

terminal

(i) X is called a miwimul m,odel if Kx is nwf; (ii) 2f there ez%sts an extremal ray R such, that the corresponding contraction. f: X 4 Y decreases the dimension, dimY < dirnX, and -Kx is ample on the genw-ic fiber F, th,en X is called a relutive Fano model. The general

following Minimal

conjecture (proved for dimX < 3) is also related to the Model Program (see Kawamata-Matsuda-Matsuki (1987)):

Conjecture 1.3.11 (ubundmce conjecture). linear Then V(X) = K(X), and the complete rn > 0 such that mKx is a Cartier divisor.

Let X be a rninimal model. system IsnKxi is free for some

It is easy to prove the following Proposition (i) (ii)

1.3.12.

Let X be a minimal

(Kawamata

(1985u))

if u(X)

there

an integer

ezists

that

iif U(X) = d’nn X, then the linear (this is an immediate consequence

Remarks 1.3.13. (i) In dimension surface, and extremal rays are only

Therefore a.fter a number either a minimal model or l?‘. (ii) For dim X > 3, category of nonsingular gularities is Q-factorial

Then:

InbKx/ # 8 for some rn,; therefore N 0; system InbKx I is free fbr some rn, > of Theorem, 1.3.5).

= 0, then

n such,

a) C is a (-1)-curve; b) C is a fiber of a @-bundle c) C is a line in p2 (see Mori

model.

nKx

0

2 one can assume that X is a nonsingular R+ [Cl, where C is one of the following:

X + r; (1982)).

of consecutive contractions of (- l)-curves, one gct,s in the sense of 1.3.10, or a geometrically ruled surface, the Minirnal Model Prograrn does not work in the projective varieties: the least admissible class of sirlterminal ones. We confine ourselves to a more general

51.4. Results on Minimal

Models

in Dimension

Three

17

class of Q-fact,orial canonical singularities. In fact, many of the theorems from the Minimal lLIode1 Program hold for the log terminal category (KawamataMatsuda-RIIatsuki (1987), Kollar et al. (1992)). The abundance conjecture is proved only in dirnension 3. (iii) Generalizations of Theorem 1.3.2 to the log terminal and log canonical cases in dirnension two and three are given in KollBr et al. (1992), KeeliVla.tsuki-McKernan (1994). 1.3.11 (iv) Kawarnata (1985b) .sh owed tha,t the assertion from Conjecture that the linear system ImKx/ is base point free for m > 0 follows from the equality u(X) = K(X), the converse is obvious. (v) From 1.3.8 (l’vIori (1982)) it follows that if -Kx is ample, then the cone NE(X) = NE(X) 1s a finitely generat,ed polyhedron every edge of which is an extremal ray.

51.4. Results on Minimal The following Theorem a smooth

theorem 1.4.1

o,f some

sequence

riety

can

X

one

(ii)

is Q-f ac t or-id X’

is a minimal

X’

Zs a

Q-fUC2 oriul there

models

algebraic contractions

mapped

onto

at m,ost

terminal

in dirnension

variety and

over

flip

3). Let X be Then by meum

@.

transformations,

a p,rojective

varieky

the

X’

belongsing

uato

classes:

is,

In case (ii)

of minirnal

of divisorial

be birationully

Three

in lVIori (1988)

three-dimensional

of the following

(i) X’

was proved

(existence

projective

Models in Dimension

with

model, relative exists

an

singularities,

or Funo model with, extremal

K.,yt is nef,

and

at most terminul

f

contraction

of one

that

singulurities.

of the following

types:

a) f: X’ b) C)

-

Y is a conic

most

rational

f:

+

X’

bun,dle

oue’r

a snormal

Y is a de1 Pezzo fibration de1 Pezzo surface) over + point; here X’ is a, Q-Fun0

(that

is,

nonsingular

a sm,ooth

f:

variety

X’

projectCue

surfuce

Y

with

at

singulardies;

the general curve with

jiber

F of f is a

Y;

Pic(X’)

= Z.

Remurks 1.4.2 (i) A, s was shown in Kawamat,a-R/Iatsuki (1987), every birat,ional class of three-dimensional varieties of general type contains only finitely many minimal models. equivalent (ii) In case a) the extrcrnal contraction f: X’ i Y is birationally (as a bundle) to a “standard form”, t,hat is, to a conic bundle f’: X” + Y’ with nonsingular X” and Y’ which is also an ext,remal contraction (see, for example, Sarkisov (198(l), Sarkisov (1982)). (iii) Some particular results on standard forrns for case b) were obtained in Corti (1996).

Chapter

18

1. Preliminaries

(iv) In case c) there is a complete list (up to deformations) of nonsingular three-dimensional Fano varieties (see also Chap. 12 of this survey). For QFano varieties with terminal singularities and p = 1, Y. Kawamata proved that the degree (-Kx) ’ is bounded by an absolute constant, from which it follows that the number of families of these varieties is also bounded. The classification of extremal rays on nonsingular varieties was obtained by Mori (1982).

projective

three-dimensional

Theorem

1.4.3 (Mori (1982), Koll&r (1994)). Let X be a smooth threeprojective variety. Let R be an, extremal ray on X and let p: X + Y be the corresponding extremal contraction,. Then only the following cases are possible: dimensional

(i) R is not numerically tion of an irreducible point. In addition, p reduced structure). All occur in this situation length, of the extremal R is a rational curve)), p(R) and [In] = R. Type

(ii)

of R

effective. Then p: X + Y is a divisorial contracexceptional divisor E c X onto a curve or a is the blow-up of the subvariety q(E) (with the the possible types of extremal rays R which can are listed in the following table, where p(R) is the ray R (that is, the number min{ (-Kx) .C / C E and 1R is a rational curw such that -Kx . 1R =

‘p and E

P(R)

lR

El

p(E) is a smooth curve, and Y is a smooth variety

1

a fiber of a ruled face E

E2

p(E) is a point, Y is a smooth variety, E E P2 and OE(E) N Q-1)

2

a line on E e P2

E3

q(E) is an ordinary double point, E = P1 x P1 and O,(E) = OP1 xP1 (-1, -1)

1

s

E4

p(E) is a double (cDV)point, E is a quadric cone in P”, and OE(E) = (3~ @ 0~3(-1)

1

a ruling

E5

p(E) is a quadruple nonGorenstein point on Y, E 2 P2 and O,(E) y Qp2(-2)

1

a line on E N IP2

x

R is numerically effective. Then ‘p: X + Y is a relative nonsingular, dimY 5 2, and all the possible situation,s a) dim Y = 2

P1 or P1 x

t

sur-

in E.

of cone E

Funo model, Y is are the following:

Results on Minimal

$1.4.

Type

of R

Models

in Dimension

cp

Cl

p has a singular

c2

cp is a smooth

Three

19

P(R)

lR

fiber

1

an irreducible component of a reducible or multiple fiber

morphism

2

a fiber

b) dimY=l Type

of R

p

P(R)

lR

Dl

the general fiber de1 Pezzo surface d,l $(dim X dimension < d, then, dim Corollary non-trivial

fiber

X ray

see (Ionescu

(1986),

be a nonsingular of length

p(R).

p =

be the set of points + /L -

1).

Wisniewski

Kawa-

projective variety. Let R c Let ‘p: X + Y be its con-

where

Moreover,

(1991b),

‘p is not

if cp has

an

isomorphism.

a non-trivial

jiber

of

E > dirn X + p ~ d - 1.

1.4.7. linder the conditions of the preceding of the morphism cp, then dim F > p - 1.

Corollary 1.4.8. There fibers of dimension < 1.

do not

exist

small

extremal

if F

theorem,

contracbions

is a

with

Now we would like to discuss briefly some new operations in the Minirnal Model Program, flips and flops. In the contraction Theorem 1.3.9 (ii) b), the exceptional locus is not a divisor, and Y is not Q-Gorenstein. Therefore even the notion of an extremal ray is not defined for Y, and the subsequent contraction process is impossible. In such a case Reid (1983a) suggested to use new birational operations (if they exist) which would be isomorphisms outside the “bad” extremal ray, destroy this “ba.d” extrernal ray and lead to a new Q-factorial variety aga.in with terminal singularities. These are the so-called flips. Definition

f:

Y

1.4.9 be a small

at

most

(Clemens-Kollar-Mori

(1988),

Mori

(1988)).

Let

llariety singularities. Let E c X be the exceptional locus, codim E > 2. A variety X+ together with a birational morphism f+: X+ + Y is called a flip if X+ is Q-factorial with at most terminal singularities, f f is an Gomorphism, in codimension 1, and K X+ is relatively f +-ample (note that, by the definition of an extremal ray, -Kx is relatively f -am,ple, that is, the operation is not symmetric). It is clea*r that the rational map (f+)-‘0 f: X- - +X+ in the following diagram X

X

+

with,

extremal

contraction

on

o, Q-factorial

projective

terminal

x----*x+ (1.4.1)

A/;: Y is

an,

codim

isomo,rphism

E+

> 2.

outside

E

c

X,

and

E+

=

(f+)-l f

(E)

c

X+,

51.4.

Results

on Minimal

Comment 1.4.10. If the flip it has an ext,remal ray generated E+. Therefore the contraction the problem of existence of flips (1988). Earlier Shokurov (1985) flips terminates. The terrnination Matsuda-Matsuki (1987).

Models

in Dimension

Three

21

exists, then either X+ is a minimal model or by some numerical class of curves out,side process can be continued. In the general case is open. In dimension 3 it was solved by Mori proved in dimension 3 that any sequence of in dirnension 4 was proved in Kawarnat,a-

V. Shokurov (1992) proved the existence of flips in a rnorc general sit,uation: for three-dimensional log terminal varieties. The terrnination in this case was proved by Kawarnata (1992c). F ram Shokurov’s proof one ca,n get a new proof of the Tori theorern (Mori (1988)). Both approaches were t,horoughly studied in (Kollar et al. (1992)), in particular, a new proof of the existence of log flips in dirnension 3 was found. Example 1.4.11 (of a three-dimensional flip). This construction was suggested by Clemens-Kollar-Tori (1988), Kollar (1991). Let F c I?” be the irnage of a geometrically ruled surface IFi under the ernbedding determined by the linear system 1s + 2fl, where s is the exceptiona,l section wit,h s2 = -1, and f is the class of a fiber of the projection Fr ---f pl. Let Y c p be the cone over F with vert,cx at P E Y. We have KF = -2s -3f, and 0~( 1) = s+2f: so KF is not a ra.tional multiple of UF( l), and therefore @(KY) @ Pit(Y) ~0 Q. Consider the composite of the nat,ural projections Y - P + IFI 4 pL, and let Xf c Y x @ c p’ x p1 be the closed gra.ph of the resulting rational map Y ~ P - - + pl. Then the hirational rnap f+: X+ ~ ~ + Y possesses the following propert,irs: (i) X+ (ii) E+ (iii)

is a nonsingular three-dimensional variety; = (f+)-‘(P) e i?)l with t,he normal sheaf isomorphic

Qlpl C-2); :X+-E+ 4 Y-P is an isomorphism f+l x+-E+ formula) Kx+ . E+ = 1, that is, Kx+ is relatively

to 0~1(-1)

#

and (by t,he adjunction ff-ample.

Next, we blow up curve E=+ on X+ and after that the exceptional section on the exceptiona,l divisor of this blow-up isomorphic to IFi. The exceptional divisor of the last blow-up is a surface isomorphic to Fc = pi x P1 which, as one can easily see, can be contracted in anot,her direction (in comparison with the last blow-up) onto a nonsingular curve E’ in a, nonsingular variety X’. The image in X’ of the first exceptiona. surface IF1 is isomorphic to p2 wit,h normal sheaf opz (-2). into a singular N ow this l?)” can be contracted point as an extrcmal ray of type E5 on the variety X. The image of E’ in X is a curve E z i?l, and, by t,he atl,junction formula, (T*Kx = Kxf + ip”. Hence Kx t E = $. One can show t,hat E generates in X an extremal ray, and E is the only irreducible curve belonging to this extremal ray. The small contraction of this extremal ray leads to the initial va.riety Y which is not Q-Gorrnstein.

22

Chapter

1. Preliminaries

Here t,he variety X is not Gorenstein. This is the general small extremal contractions on a Gorenstein three-dimensional (1988)) Clemens-Kollar-Mori (1988)).

fact: there variety

are no (Mori

Remark 1.4.12. This example shows that it is rather difficult to const,ruct, examples of flips even in the least possible dimension 3. That is why the approach due to Mori (1988) to the proof of the existence of flips seems to be very important. It is reduced essentially to their classification (Kollar-Mori (1995)). The remarkable fact noticed for the first time by Shokurov (1985) is that singularities of X+ are simpler than those of X. The more simple birational transformation which is also an isomorphism in codimension 1 is a flop. We confine ourselves to the three-dimensional case. Definition 1.4.13 (see Reid (1983a), Kollar (1989), Kollar (1991)). Let X be a Q-factorial projective three-dimensional variety with at most term% nal singularities, and let D E Div(X). A bl;rational map X: X ~ + X’ is culled a D-flop Zf there exists the folloukg commutative diagram:

where (i) (ii)

(iii)

(iv)

f and f’ are birational

morphisms

(X = f’-’

0 f) such that:

X’ is a Q-factorial projective three-dimensional variety with at most terminal .singularities; let E c X and El c X’ be the exceptionzal loci for f and f’ respectively; then codirn E 2 2, codim E’ > 2, and X: X ~ E + X’ - E’ is an isomorphism; the morphisms f and f’ cannot be decomposed into compositions X + Yl + Y and X’ --) Yl 4 Y in such, a way that conditions pi) and (ii) for Y, would be satisfied; if D’ = X*(D) is the proper transform of the divisor D, th,en -D is relatively f-ample, and D’ is relutillely f’-am,ple. Consider

now the following

sirnplest

Example 1.4.14. Let 2 he a nonsingular three-dimensional complete variety containing a surface F 2 IP1 x IPi with normal sheaf NFIz F UP1 xPl (-1, -1). Let Y be the variety obtained from Z by contracting F into an ordinary double singularity. Assume that in the commut,ative diagram

g2.1. Definitions,

Exarnplcs

and Simplest

Properties

23

the varieties X and X’ obtained by contracting F onto nonsingular X and E’ C X’ along two pencils of lines are projective. Then is a flop transformation. Theorem 1.4.15 (Reid (1983a); Kolllir (1989), Kollar f: X + Y, D und E be as in Definition 1.4.13. Then: (i) (ii) (iii)

curves E c X; X - - +X’

(1991)).

Let

a D-flop exists; every sequence Xi: Xi- ~-X: of D;-flops is finite, where (Xi-l)*D,-l is the proper tmn,.sform, o,f D7-l, i = 1,2, . . . . if X is nonsingular, then X’ is also nonsingular.

Di

=

Remarks 1.4.16. (i) The proof of Theorem 1.4.15 in KollSr (1989) is based on an explicit construction. The first, proof is due to Reid (1983a). (ii) Flops do not change the nature of singularities in the sense that analytic neighborhoods of the corresponding points are isomorphic. (iii) If X is nonsingular, and p(X) = 2; then a D-flop does not depend on the choice of divisor D, becwse Kx . E = Kxl E’ = 0. (iv) Flops are applied extensively in three-dimensional hirational geometry. For example, a minimal model of a three-dimensional variety X of genera,1 type with Q-factorial singularities in general is not uniquely deterrnined, and any two of them are linked by a birat,ional transformation which factors into a composition of flops (Kollar (1989)).

Chapter 2 Basic Properties of Fano Varieties $2.1. Definitions,

Examples and the Simplest Properties

Definition 2.1.1. A smooth projective *variety X is called a Fano uurieey if its an,ticanonicul divisor -Kx is ample. If u normal projective variety X has sin,gulur points (for example, terminal, canonical, etc.), and some positive integral multiple -nKx, n E W, of an, anticanonicul We41 di*visor -Kx is un am,ple Cartier divisor (that is, in particular X is Q-Gorenstein), then X is culled a sin,gulur Funo uuriety. A Fano variety with terminal Q-fuctoriul singularities and p = 1 is called a. Q-Fun0 variety. Finally, a log terminal pair (X, B) is culled u log Funo variety if a @Curtier divisor Kx + B is ample (that is; some its posikue mulkiple being u Cartier divisor is um,ple). If B = 0, then X is simply said to be a log FanTovariety. Proposition resolution,

2.1.2. of skgulwities.

Let X Then:

be a log Furl0

variety,

and

let

f:

Y +

X be u

24

Chapter

2. Basic Properties

of Fano Varieties

(i) H”(X, (3~) = Hi(Y,Oy) = 0, vi > 0; (ii) Pit(X) and Pit(Y) are finitely generated torsion-free Z-modules; Pit(X) N H”(X, Z), and Pit(Y) ‘v H”(Y, Z); (iii) numer-ical equivalence on the set of Cartier divisors on X, respectively on Y, coincides with linear equivalence; (iv) 6(Y) = -cm. Assertion (i) follows from the Kawamata-Viehweg vanishing tkleorcm (see 1.3.4). The second part of assertion (ii) follows from tile cohomology sequences for exact exponential sequences of sheaves over @.

assertion (i) and the isomorphisms Pit(X) y H1(X,O>), Pit(Y) N Hl(Y. 0;). To prove (iii), one uses the following consideration: if D and D’ are Cartier divisors on X and Y respectively, and D = 0, D’ = 0, then from the Riemam-Rock1 theorem and the Kawamata-Viehweg vaniskling theorem we get H”(X, 0x(D)) = H”(X,Qx) = 1, i?(Y,c?y(D’)) = H”(Y, 0~) = 1. Since X and Y are projective, it follows from the above that D = 0 and D’ = 0, and that Pit(X) and Pit(Y) arc torsion-free. Corollary See (Koll&r acteristic.

2.1.3. (1996a))

Let X be a log de1 Pezzo f OI. g eneralizations

surface.

Then

X is rational.

of 2.1.2 a,nd 2.1.3 to positive

char-

Remarks 21.4. (i) C orollary 2.1.3 cannot be generalized to Fano varieties of dimension > 3 even in the nonsingular case (see Chap. 8). (ii) From 2.1.2 (ii) and Lemrna 1.1 from (Kawarnata (1988)), it follows that the class group of Q-Weil divisors Cl ~i3Q is finitely generated. But) unlike the case of Pit(X), the group Cl(X) ca.n have torsion. One can find such examples among those three-dimensional variet,ies X c IPN whose hyperplane sections are Enriques surfaces (see Cont,e (1982)) Conte-Murre (1985)). In view of 2.1.2 (ii), for a log Fano variety X there exists the greatest Cartier rational r = r(X) > 0 such that -K x = rH for some (ample) divisor H (the equality klere means the equality of elements of the group Pit(X) @ Q); such an r = r(X) is called the index of the Fano variety X, and the divisor H (respectively the linear systern IHI) is called a fundamental divisor (respectively the fundamental system ) on X. We shall call the selfintersection index d = d(X) = Hdir’lX tkle degree of tkle Fano variety X (if H = -Kx, and -Kx is very ample, then d is the usual degree of X with We remark respect to the anticanonical embedding cpipox I: X pi ~‘*irrll~KxI). that if X is Gorenstein, then the index r = r(X) is a positive integer.

g2.1.

Definitions,

Examples

and

Simplest

Properties

25

Examples 2.1.5. (i) In d’unension 1 there is a unique Fano variety up to isomorphism. namely pl. Two-dimensional Fano varieties are called de1 Pezzo surfaces. The following is the complete list of them: p2; a smooth quadric Q c p”; a geometrically ruled surface Fi with the exceptional section s, s2 = -1; a series of surfaces F = Fd c P” of degree d = Kg, 3 5 d < 7; a double cover Fz of P” ramified along a smooth curve of degree 4; a double cover Fl of a quadric cone Q’ c p” ramified along a smooth curve of degree 6 not passing through the vertex of the cone. Surfaces Fci can be obtained by blowing up of 9 - d points on p2 which a,re in general position. In particular: all de1 Pezzo surfaces are rational (see Manin (1972), Nagata (1960) and also 2.1.3). (ii) An analogous classification exists for de1 Pezzo surfaces with ca.nonical (i.e. Du Val) singularities (see Du Val (1934), Demazure (1980), Brenton (1980), Hidaka-Watanabe (1981)). If X is a singular de1 Pezzo surface with Du Val singularities, and (-Kx)~ = d, then X is one of the following: 1) d = 8, X = Q’ c p’ is a quadric cone; 2) 3 < d < 6, X = Xd c pd is a projectively normal surface of degree d; 3) d = 2, X can be represented as a double cover X + p2 rarnified along a singular curve of degree 4; 4) d = 1: and X can be represented as a double cover X 4 Q’ of a quadric cone ramified along a singular curve cut out on Q’ by a surface of degree 3. A rninimal resolution X of the surface X in all the cases but 1) is a blow-up of p” at, 9 - a! points which are in almost general position; see Demazure (1980). (iii) Let G c P GLz(k) be a finite group, and S = p2/G. Then X is a log de1 Pezzo surface with Picard number p = 1. In particular, cones over rational normal curves can be represented in this way. Nevertheless, it is known that not every log de1 Pezzo surface with p = 1 is of the form p2/G (see AlexeevKolpakov (1988)). Log de1 P ezzo surfaces have been studied by V. A. Alekseev and V. V. Nikulin. The complete classification of log de1 Pezzo surfaces with indices of singular points 5 2 was obtained in Alexeev-Nikulin (1989). The set of possible values of indices of log de1 Pezzo surfaces was described in Alexeev (1988). The questions concerning the boundedness of the degrees of such surfaces in terms of rnuhiplicities and indices of singularities were considered in Nikulin (1989a), Nikulin (1988)) Nikulin (1989)) Alexeev (1988). (iv) Now we give some of the sirnplest examples of Fano varieties of dimension n = dimX > 3. First of all, these are projective spaces p’“, smooth hypersurfaces Xd c p7’+l of degree d 5 n + 1, smooth complete intersections X cl1,,,,,dl c pN of hypersurfaces of degrees dl, . , dl such that Cd, 5 N. Let Y be a Fano variety of index T. Let H be a fundamental divisor on Y (that is: -KY = T-H), and let 7r:X ---+ Y be a cyclic cover of degree m with smooth ramification divisor B c Y. If B N bH, where b < rm, then it follows from the Hurwitz formula Kx = n*(Ky + &B) that X is a Fano variet,y of index T’ = T - $ (note that m divides b).

26

Chapter

2.

Basic

Properties

of Fano

Varieties

A more general construction (see Tori (1975), Dolgachcv (1982)). Let p = iP(q~. , Ed) be a weighted projective spuce, that is, p = Proj ~[zo: . ,TN], where degz; = ei. Let X c p be a smooth complete intersection of hypersurfaces of degrees dl, . , d, which does not intersect the set U~>~SI;, where Sk c p is the closed subset determined by the ideal generated by {:q 1 k ,/‘el}. Then the restriction of the sheaf (3~( 1) to X is an invertible sheaf 0, (l), and if dim X = N - m 2 3, we have Pit(X) = Z. 0~ (1). The canonical class of X can be calculated by the formula Ox(Kx) = 0,(x d, - c ei). From this we find that if Cd? < C e,, then X is a Fano variety of index T = C e, - Cd+ with p= 1. (v) Grassmarmians G = Gr(‘I*L, N) of rn-dimensional subspaces in an Ndimensional linear space are also Fano varieties, and -KG = N . H, where H is a hyperplane section of G under the Pliickcr embedding G = Gr(m, N) 4 I~D(A”“~~). From t,he adjunction formula, it follows t,hat smoot,h complete intersections of G with hypersurfaces of small degrees are also Fano varieties. In part,icular; a general section of Gr(2,5) c pg by a subspacr of codimension 3 or by a subspace of codimension 2 and a quadric are three-dimensional Fano varieties X:, c p” and Xl0 c p7 with p = 1, and -Kx, = (3x, (2). -h-xx, = ~XK (1). (vi) Q-factorial Fano varieties naturally arise in t,he Mori Minimal Modr>l Program. Many examples of such varieties can be fourld among toric varieties (see below). Other examples of sing&r (in particular, Q-Fano) varieties can be obtained as quotients of (nonsingular) Fano varieties by finite group actions (in the general case these varieties have log terrninal singularities; see ClemensKoll;ir-Tori (1988)). For example, let G c PGL,+l(k) be a finite subgroup; then p”/G is a log Farm variet,y with p = 1 and Q-factorial singularities. If G c PGLd(lc) is a subgroup of order 2, then the following t,wo cases arc possible: a) G is generated by the element y = Diag(1, l,l, -1); in this case p”/G is isomorphic to t,he cone over the Veronese surface a.nd is a Q-Fan0 variety (see Example 1.1.1); b) G is generated by the element y = Diag(1, 1, -1, -1); in this case X = p”/G is isomorphic to the complete intersection of two quadrics .cz = y2 and tv = u2 in IID”. The variety X has canonical Gorenstein singularities along two tonics which are rut out by the planes z = y = z = 0 and ?I. = 71 = t = 0. (vii) If is nef and for m > ~r~oHO(X~

on a smooth projective variety X, an anticanonical divisor -Kx big, then, by Theorern 1.3.6; the linear system ( - rnKx/ is free 0. Frorn this it follows that the anticanonical ring R(-Kx) = model 0x(-mKx)) is fi m‘t e 1y gencra.ted. Then the antica.nonical

xl,,, := Proj R( -Kx) isomorphism X + X-,,,) gularities.

is hirational to X (there exists a natural birational and is a Fano variety with at most, canonical sin-

32.1. Definit,ions,

Exarnples

and Simplest

Properties

27

(viii) (T. Fujit,a) Let X b e a log Fano variety, and let -Kx = rH for some ample Cartier divisor H on X. We set, f = 0~ @ Ox(H). The linear syst,em is the t,autological linear bundle on p(I), is base PP(E) (71~11. where Qy~)(l) point, free for m, >> 0 and determines the morphism p(E) + X’ c pN whic*h contracts the exceptional section E. It is easy to check that, K$D(E) = f*Kx’

+ (r ~ l)E

From t,his we get tha,t X’ is a log Fano variety of dimension dimX + 1 and index I’ + 1. This va.riety is ca,lled a generalized cone over X. For example, let X = p”. O,(H) = C3p(d); and T = $. Tl ien X is a usual cone over t,he surface S”(~Z which is the image of p2 urlder the embedding det,ermined by the linear syst,em lUp~(d)/. If d = 1, then X’ r” p’; if crl = 2: then X’ is the cone over the Veronese surface (see the example above); if d = 3, t,hen X’ is a Fano variet,y with Gorenstein canonical singularit,ies; if d > 4, t,hen X’ is a log Fano variety. We remark that, the la.st examples can also be obtained as quotient,s of p3 by actions of abelian groups. The natural generalizations of Grassmanrlians (see Example 2.1.5 (v)) arc homogeneous spaces. Consider a semisimple 1inea.r a.lgebra.ic group G and its parabolic subgroup P c G. Then the quotient X = G/P is a projective algebraic variety which is a liomogeneous space with res1)t:c.t t,o the act,ion of the group G. Now we need to introduce some notation. Let T c I’ be a maximal torus, and E := Hom(T.@*) $Z R. Let A c E he the root system corresponding t,o G. Let C c A be a basis of the root system A (that, is, a set of simple roots): and let W = Nc;(T)/T he the Wry1 group. We can assume that 1’ = P(6) 1s . a standard parabolic subgroup, that is, P is generated by some fixed Bore1 subgroup and the group We, whcrc: We is the subgroup of W generated by reflections with respect to the element~s of 8. The cohomology ring H*(X, Z) and the total Chern class c(X) of the variety X = G/P ran be expressed in t,erms of t,he root syst,ern A (see Bore1 (1953), Borel-Hirzebruch (1958)). H err we rest,rict ourselves t,o the description of H2(X; Z) and cl(X) = -Kx. Proposition (i)

2.1.6

There exists a H2(X, Z) ---f E.

V:

invariant

(ii)

bilinear

(1953),

canonical

Borel-Hirzebruch

kjective

(1958)).

hom,om,orph,ism,

The im,age of v is orthogonal form on, E) to th,e elemuds

of

(with ai,

respect

th,at

to the

positive E ‘q@.

positive basis

roots C

at least

Q,

such

one

that

in

element

the

roots

and,

to

C\O

and

ondy

W-

co~~l,~)lerrl,en,tary

decomposition

from

the

is generated

occurs

of CE, with with

a strictly

coeficient.

A71 element Q,

is,

Z-modules

respect

i E 0,

by the fundamental weights Wi, i @ 8. q(X) = CQ~, w h ere a, runs ouer the set of positive to 8,

(iii)

(Bore1

b E

H”(X,

23) is positive

if

if

(b, 0,)

>

0 for

all

Chapter

28

Corollary ety X = G/P,

2.

Basic

Properties

of Farm

2.1.7 (Borel-Hirzebruch (1958), the anticanonical divisor -Kx

Varieties

Kollar (1996a)). On the sariis ample, that is, X is a Fano

Uariet~J.

It is also possible to describe the Tori cone NE(X) and extrema.1 rays on X in terms of subgroups of G: if P’ > P is another parabolic subgroup of is a contraction of an G, then the canonical morphism X = G/P + G/P’ cxtremal face, and every contraction of an extremal face is of this kind; in particular, every extremal ray on X is numerically effective. In the case G = SL,, (c), the variety X = G/P is a va.riet,y of flags in Y?” -‘. Example %.1.8. Let G = SLa(c), and let P = B be a Bore1 subgroup (the subgroup of upper triangular matrices). Then X = G/B is the variety of complete flags in p2, and dim X = 3. The root system A is of type AZ (sw Fig. 2.1.8), C = { 01, QZ}, 0 = 8. The group H2(X,iZ) is generated by the

Fig.

fundamental Calculate ical divisor

weights Wi = $oi + ~QS, 7& = ioi + icu2; therefore p(X) = 2. now the index T of the Fano variety X. The class of an a.nticanori-A-x is the surn of all positive roots:

q(X) Thus rays, G2). p2 x (see

1.

= al+

Q2 + (cl1 + Q2) = 2(Qi

+ Q’L) = 2(211i + WIz) .

we get that T = 2. Since p(X) = 2, t,hen X has exactly two extremal (that is, are of type and their contractions ‘pl: X + p2 are IF’l-bundles The variety X can also be realized as a divisor of bidegree (1,l) on p2 or as the project,ivizat,ion ~?~z(T~z) of t,he tangent bundle Tpa of p” 3.3.1).

Consider now homogeneous varieties X = G/P with Picard number p(X) = 1. In this case the parabolic subgroup I’ should be maximal, and the corresponding subset, 0 c C is obta.ined from C by removing one element. Therefore we ca,n a.ssume the group G is simple. We get the following examples.

fj2.1.

Definitions.

Examples

and

Simplest

Properties

29

Examples 2.1.9 (Borel-Hirzebruch (1958)). (i) G = SL,,, (C). Up to conjugation, G contains exactly m - 1 maximal parabolic subgroups Pi, . . . , l&i. The quotient varieties G/Pk are isomorphic to the Grassrnannians Gr(k, m). (ii) G = SO,,,(@). F or every maximal parabolic subgroup P c G, the quotient variety G/P is isomorphic to the isotropic Gra.ssmannian, that is, to the subvariety of Gr(k, ,m) consisting of k-dimensional linear subspaces symmetric V c @“’ such that q(V, V) = 0, where q = (., .) is a non-degenerate bilinear form on P. Consider now the cases of small nb. a) m = 5. The root

system

n is of type B2:

One can assume that P = I?, = Pe, , where 0, = n - {ai}. We get two varieties: G/PI which is isomorphic to a smooth three-dimensional quadric Q” c P4; and G/P 2 which parametrizes lines lying on a smooth t,hree-dimensional quadric (it is known that it is isomorphic to p’). b) m, = 7. The root system n is of type &:

Similarly,

we get three

varieties: quadric Q” c P”; variety parametrizing lines on Q” c IP”; variety parametrizing 2-pla,nes on Q” c IP”. roots complementary to 02 = n - {oz} is equal to

G/PI, a 5-dimensional G/Pz, a 7-dimensional G/Pz, a 6-dimensiona, The

sum of positive

4(o!i + 2Q2 + 2a3) = 4702 ) that is, G/P, is a 7-dirnensional Fano variety of index 4. Likewise, we get that G/P3 is a 6-dirnensional Fano variety of index 6 (and therefore G/P3 is isomorphic to a quadric Q” c P7, see X1.15). (iii) G = Sp,(@). If P c G is a maximal parabolic subgroup, then G/P is isomorphic to the isotropic Grassrnarmian with respect to a non-degenerate alternating bilinear form. (iv) G is a group of type G2. There exist exactly two rnaxirnal parabolic subgroups PI and Pz in G. The quotient variety G/PI is isomorphic to a 5dimensional quadric Q” C P”, and G/P2 is a 5-dimensional Fano variety of index 3 (see Mukai (1989)). A sirnilar description can also be obta,ined type F4, Eo, ET or E8.

for G/P,

where

G is a group

of

30

Chapter

2. Basic

Properties

of Fano

Varieties

Another convenient way to const,ruct many examples given by toric geometry (see Kempf et al (1973), Danilov Definition. un

ulgebraic

the

Zariski

of Fano varieties is (1975), Oda (1978)).

A variety X is called toric if them mists cm c&ion, T : X of T on X by biregulur untomorphisms which hus u dense (in topology) orbit U c X equivukvntly isomorphic to T. torw

Let M = ZY. Let N = M* be t,he dual Z-module, and let Nx = N @ R. Elements ~1 E M, rn, = (ml, . , m,,), correspond to monomials :?” = 2-1ml . ..x., 7’1r1 t C[M] = qq,. .) XI,,]. To every cone g c NR (we assume t,hat, all the cones are determined by a finite number of inecmalities of type rn, 2 0, where rrb l Al 8 R), one can associate a finitely generated ring c.[3 n M] a.nd, therefore, an afhne toric variety X, = Spcc@.[c? n Al]. A fun F is defined to be a firme set of cones c c Nn such that the following three conditions hold: 1) if g E F and r E F, then 0 n 7 E F; 2) if 0 E F and ‘T is a face of 0, t,hen 7 E F; 3) every cone a E F has a vertex. For every fan F, t,here is a. t,oric variety X&. associated t,o it; XF is covered by affinr toric varieties X,; where (7 E F, and every two vari&ies X, a.nd XA are glued along X, if 7 is a r~~nrrm~ face of (T and 0’. It, is known that rver,y toric variet,y can be obtained in t,his way. We consider here only complete varieties XF. This condition is satisfied if and only if the fan F is cornplet,c, that is: NR = U (T (see Danilov (1975)). All t,he inforrnat~ion cornerning the va,riety Xp can be cxprcssetl in terms of t,he lattice N and the fan F. For example, the Picard group Pic(XF) is isomorphic to the quotient group of the group of piecewise linear functions on Njk having integer values on N by the subgroup M (see Danilov (1975), Oda (1978)). Examples 21.1 II. (i) Let M = Z”, N = M*. Let ei: , c,, be a. basis in N, and q) = -ei - e,,. Consider the fan F consisting of cones (ql. i et,), Ic < 71, 0 < i, < n,. Then the variety XF is isomorphic to p”. It is known that, every smooth complete toric varirty with p = 1 is isomorphic to p’“. (ii) Toric de1 Pezzo surfaces are exactly the following ones: p2; p’ x p’, IFi j Ffj, F7 (see Example 2.1.5 (i)). (iii) The complet,e classificat,ion of t,hree-dimensional toric Fano varirt,ies was obt,ained in papers of Batyrev (1981), Watanabe-Watanabe (1982) (see Table 12.8. Chap. 12). Every such variety Xf,~ is determined by a fan F c NR which in its t,urn is determined by a Z”-weighted triangulation of the two-tlirrlensiorlal sphere S” c R” = NR (every vertex of the triangula.tion corresponds to a ray in F and its @-weight corresponds to coordinat,es of a primitive vector from N lying on this ray). There is an upper bound p 5 5 for the Picard number of tllree-dirnerlsional t,oric Fano va.riet,ies. In the mse p = 2 the triangulation of S2 c IR” is determined by the graph represented

52.1. Dcfinitiorls.

Examples

and Simplest

31

Properties

Fig. 2. on Fig. 2.1.10, and @-weights are determined has only the following possibilities:

1) N=

(

1 0 0

0 1 0

is the blow-up

0 0 1

-1 -1 0

-1 -1 1 ) -1

by the matrix

x

= Pp(Op1

N for which

0 op

one

:I) c+(l))

of a lint: on i?“, 7’ = 1, 9 = 28;

2) N =

1 0 0

0 1 0

0 0 1

-1 -1 0

0 0 -1

01 0 010-l

01

-12

-10 0

3) ( Iv=

X=P”xIP1,

)

7.-l,

!/=28;

x = IPp (Opz G>op (2))

i is a I?“-bundle

over P”, 1’ = 1; 9 = 32;

4) Iv=

1 0 010-l ( 0 0

is a the blow-up

0

-1

1

1

0 0 ) -1 1

of a point

x = Pp(Op

CD&a(l))

on IP, fir= 2, cl = 7.

Remarks 2.1.11. (i) The list of four-dimensional smooth toric Fano varieties can be found in Batyrev (1984) and Batyrev (1998). classified all smooth toric (ii) Voskresenskii-Klyachko (1984) completely Fano varieties X which possess the following property: the involution 7: T + T, t + -t of the torus T can be ext,ended to a biregular involution of the variety X. Model Program holds for toric (iii) Ried (1983 c,) 5.1-lowed t,hat the Minimal varieties of any dimension with terrninal Q-factorial singularities (see also Danilov (1982)).

Chapter

32

2. Basic

Properties

of Fano

Varieties

(iv) Toric geornetry can be used to construct a great number of examples of singular Fano variet,ies. For example, weighted projective spaces a.nd any quotients of lY” modulo abelian groups are toric Fano varieties. A. A. Borisov and L. A. Borisov obtained the following result: every three-dimensional toric Q-Fan0 variety is either a weighted projective space of one of the following types p( 1, 1, 1, 1) = p” ~ p(2, 1 i 1,1) (the cone over the Veronese surface), p(3,2: 1, l), E?(5,3,2, l), p(5,4,3, l), i?(7,5,3,2), l?(7,5,4,3), or a toric variety deterrnined by t,he simplex in N with vertices at, points (l,O, 1): (-2,1, l), (1, -2,O), (0, 1, -2). Next we discuss The proposition vanishing theorem.

the simplest, properties of Fano varieties. below follows immediately from the Kawamata-Viehweg

Proposition 2.1.12. Let X be a n-dimensional dex r, and H be 0, fundamental di,visor. Then H”(X,

CIx(mH))

Corollary 2.1.13 (Shokurov does not exceed dim X + 1.

V’i > 0,

= 0, (1985)).

lo9 Fano

variety

of in-

‘drn > -r

Th e index

of a log Fano

variety

X

Proof. By the Riernam--Roth theorem, y(Ux (mH)) is a polynomial of degree dimX. By Proposition 2.1.12, the roots of this polynomial are integers n=-l.-2:...,1-prl,whencer n - 2, then H”(X,O,y(H))=

(ii)

andd=lforr=n+l,d=2forr=n,; if r = n - 2, an,d X has at most H”(X,&(H))

id(r-n+S)+sr-1,

canonical

Goren,stein

singularities,

then

=g+r~-1,

where y = 2 + 1 > 2 is an integer The integer g = g(X) = q + 1 is called the gfzn?Ls of an n-dimensional variety X of index n ~ 2. It can also be defined for a three-dimensional variety X with Gorenstein singularities of arbitrary index to be ~(-Kx)”

Fano Fano + 1

The reason for this definition is the fact that for a three-dimensioial (norisingu1a.r) Fano variety X with very ample -Kx, the image ‘pl-~~ I (X) = X 29 -I 2 c p”+’ is a variety of degree 2g - 2, the sections of which by subspaces

$2.1. Definitions,

Examples

and Simplest

Properties

33

of codimension 2 are canonical curves of genus g. The converse is also t,rue, that is, every smooth three-dimensional variety Xzyp2 in p”+l with canonical curve-sections is a Fano variety (see ContIe-Murre (1986)). In the original papers by G. Fano, this property was taken to be the definition of the cla.ss of varieties urlder investigation. The ample sheaf 0~ (H) 1s not necessarily generat,ed by its sect,ions; that is, the corresponding mapping ‘piHi: X - - -l?’ dir” lHl is not, necessarily a morphism (see Sect. 2.3). For three-dirnensional Fano variet,ies X with (~1~1 being a morphism, we have the following Proposition 2.1.15. Let X be a three-dimensional Fano variety of inderr with canonical Gownstein singularities, and let -Kx = rH, where H is a fundamental divisor on, X. Assume that the linear system 1H 1 is base point free = and determines a morphism ‘~1~1: X 4 yl~l(X) C p dirrllH1. Then degplHl 1 or2. Proof.

Let d = H”.

Substituting

formulas

degplHi(X)

2.1.14 into

> codimplHi(X)

the inequality

+ 1,

which is obviously true for any subvariety of a projective space not, lying hyperplane (set Corollary 3.1.4)) we obtain the inequalities: deg:cPI~~~

d

>-

ldr 2

29 - 2 deg ‘PIHI

>

g - 1

- 1

where g = g(X) is the genus of the Fano possible only when degplHl = 1 or 2.

in a

for r > 2 ;

for 1’ = 1 i variety

X.

These

inequalities

are

The complete classification of nonsingular three-dimensional Farm variet,irs for which the anticanonical linear system gives a finite morphism of degree 2 onto its image (in Iskovskikh (1977) such varieties were called hyperclliptic) (1979a), Iskovskikh was obtained by Iskovskikh (1977) (,see, a 1.so Iskovskikh (1988)). Theorem 2.1.16 (Iskovskikh (1979a)). Let X be a nonsingular threedimensional Funo vuriety of in,dex r and germs y. Assume th,ut the untican,on,ical linear system j - Kx j determines a morphism ‘p: X + X’ C iF+’ uhich ramiis not un embedding. Then qx X + X’ is a double cover with a smooth fication divisor D c X’. The uuriety X is completely deterrrked by the puir (X’, D), and for this puir only the followin,g cases are possible: (i) X’ c Y” *is a cone over the Veronese surface, an,d D c X’ is cut out by a cubic hypersurface; in this case X is isomorphic to the vuriety from 3.1.6

(i);

34

Chapter

(ii)

X’

(iii)

Pit(X)

X’

= Q

Properties

of Fano

und D c p3 is a surfc~ce of de,9ree 6;

= p’.

and

2. Basic

Varieties

this

in

case

r = 1, g(X)

= 2,

= Z;

c I? ’ is u smooth

and D c Q is cut out by a quadrk

quad%,

P”; in this caser = 1, g(X) = 3, and Pit(X) (iv) X’ = p’ x p2 c p5 embedded (2,4); in th*is case r = 1, g(X)

in

= Z;

by Segre, and D is a divisor of bidegree 4, and Pit(X) = Z @Z; (v) X’ = ppl (E), where E = OF1 (2) q; C&M (1) 33 C&l (1)) and X’ is embedded in IP” by the linear system IO~(EJ (1)1, and D E IO,(s) (4)1; i7~ this case r = 1, g(X) = 5, and Pit(X) = Z CDZ; CILe variety X can also be real&d as a blow-up of a Fano variety YL of &de: 2 ulon9 a nonsingulnr elliptic curve HI n Hz, where HI, H2 E $1 - KY> / ; (vi) X’ = p1 x j?” c p” embedded by the lkr,ea,r system IpTC$(2) 8% p$c?p~(l)J~ and D t j&C& (4)1; %r~ this case r = 1, g(X) = 7, and Pit(X) = Z”: th,e uuriety X is isomorphic to P1 x F7 wh,ere F is a de1 Pezzo surfare of degree 2.

Corollary

=

2.1.17 (cf. Iskovskikh (1979a)). Let

Fano ‘variety of index r = the fundamental linear system, I - h Kx p:x + X’ c jp+rr-2 which is not an double co’ver w& a smooth rumi,fication completely determined by the pair (X’, D). the following possibl,litaes. n,onsingular

X

be

an

v-dimension&

an,d gerrus g. Assume that / = IHl determines a morphism em,bedding. Then p: X + X’ is a disvisor D c X’. The variety X is If n > 4, on,e has for this pair only n ~

2

(i) X’ = iF”, and D is a hypersurface of degree 6. In this case g(X) = 2, and Pit(X) = Z. (ii) X’ = Q C lF+1 is a quadric? and D c X’ is C?L~ out by a h:ypcrsurfuce of de,yree 4. ln this ca,se g(X) = 3, and Pit(X) = Z. (iii)

X’ n

= P1 x p” c IED embedded =

3:

g(X)

=

by Se,yre, and D E lpsO,:l(4)1. In this cuse = Z @ Z. The variety X is Gomorphic Yl is a three-drmensional Fano variety of index 2 and

5,

and

to p1 x Y2, where degree 2.

Pit(X)

$2.2. Some General R,csults Definition

2.2.1. A norm,al aariety Y c PN which does 71ot (it is often said to be non-degenerate) is called projectively of the following clearly equivulent con,ditions is sntisfied:

l%e

perplane one

(i) for

any

C+(m)

(ii)

the

naturd

*Lnt eg esr ‘rn on, Y is cut

> out

homomorphism

1, eueyy divisor from

the complete linear by some hypersurface of degree rn %n PN; of graded algebraCs

S* H”(Oy(l))

--) ,JE,~H~(W)) -

in

a hy-

n,ormal

i,f

system

$2.2.

is surjective,

,where

Sorne General

Results

35

Ho(.) is the symm,etric algebra generated

S”

hrj

the

.spuceHO(.). Definition 2.2.2. A variet?J Y c pN is called fine if it is projectively and h,“(Oy (m)) = 0 for all i, 0 < i < dim Y.

normal

Let Y be an irredwible complete variety. Let L be an ine a non-zero se&on, an,d let D c Y vertible sheaf on Y. Let s E H”(Y, L) b be the dikor of zeros of the section s. Assume that: Lemma

(9

(ii)

2.2.3.

H”(Y. L) + H”( D, LID) is swjective; H”(Y, L) --) HO(Y, Lc4(7r1p1))are surject%,ue for power of m > 2, where ,‘PP1 Ho(.) denotes tl beI ( m ~ 1)-st symmetric HO(.). Let I(Y, L) be th,e homogeneous ideal which is the kernel of the homomorphism, S* H”(Y, L) + c$J,~~)~~H~‘(Y, LBTrL), and let I,,,(Y. L) C I(Y, L) be its homogeneous part of de,greem. Similarly we define I(D, LID) and I,,,(L). LID). the

natural

the

h,omom,orphism,s

homomorphism

SrrLP1

Then for any element F,,, c I,,,(D, LID), there exists un element F,,, E that is, the horrrorn,orph~s’rrL Q,,, : I,,, (Y, L) + I,,,(Y, L) such that F,,, 1D = F,,, , I,,, (0, Lj D) is surjective; moreover, F2 (for m = 2) is ~uniquely deterrnisned !ITJ the element Fz (Note that the con,ditions of the lemma are obviously sut*i.sjied for projectively normal vurieties Y c PN and L = C&(l) with any rn > 0). Proof. Consider

the following

commutative

0 I

0 J.

I,,,-1(Y, L)

O--t

diagram:

J%

0 I

Im(Y,L)

I,,, (D, LIDI A

=

I P--l

o-

I&L) .l H”(y, L@‘f-1))

o-

2 -

S”’ H”(Y.L) i I H”(Y, LRTn) -

S”’ H”(D, LID)

+

0

H”(L). (&)

0 where the exact,ness of t,he first column is prc>ciscly condition (ii), and the exactness of t,he second row is condition (i) of the lemma. The desired surject,ivity of the homomorphism Q,,, can be proved by the standard diagram method (the snake lemma.). The urliqueness of F2 is obvious. Corollary

2.2.4

assumptkns

of

normal

Y

integer

r,

ifDcp %nterse&on

c

PN),

if

I(D) is also

is an

intersection

of

quudrics

for all is generated

2.2.,9

I(Y)

then NP’

(see, for example,

Lem,ma

gen,erated

containing

Iskovskikh m

>

by

2

(1977)).

(for

elements

by elem,ents

Un,der for

the

prqjectively

of degree < r ,for sorn,e of degree5 I’ In particular,

of qmdr1c.s containing Y.

example,

D,

th,en

Y

is also

an

36

Chapter

Lemma quence

2.2.5.

2. Basic

Properties

In the preceding HO(Y, L@y

of Farw

notation,

+ H”(D,

Varieties

assume

(L\D)@“‘)

that th,e restriction

se-

4 0

is exact ,for all m > 1. Then if th,e graded algebra &>oH0(D7 (LID)“““) is generated by elements of degree < r for some integer r, then the graded algebra by elemesnts of degree < r. @-7,z>oH0(Y, L@““) is also generated This

is proved

by induction

as well as the preceding

Lemma 2.2.6 (Iskovskikh genus g, and let L = C+(D) Then the sequence

(1979a.)). Let be an invertible

0 + I(Y, L) + S* H”(Y,

L) +

corollary.

Y be a nonsingular curve of sh,eaf of degree deg L > 2g + 1.

@ HO(Y, LNrrr) 111 >o

+ 0

is exact, and the ideal I(Y, L) is generated by elements of degree 2 in, all the cases except for the case when g = 1,deg L = 3, that is, when, Y c p2 is a cubic curve. Lemma 2.2.7 sheaf on a smooth

(Iskovskikh (1979a)). Let L = O(D) variety Y. Assume that the map S”’ H”(Y,

L) --f H”(Y,

be an ample

L@“)

is surjective for all m > 1. Th,en divisor D is very arrbple PlDi (Y) C pN is obviously projectkuely normal). Proof.

Consider

the following

commutative

43

‘-4

~(H”(Y

(and

the image

diagram:

P(Srrr H”(Y,

Y \

invertible

L))

L))

Here (~l,,~~l is an embedding for some m > 0: Q is an embedding by the assumption of the lemma that the map is surjective; /3 is the standard Veronese embedding. Frorn this it follows that ‘piul is also an ernbedding. Corollary 2.2.8. Under the condition,s of Lemrna 2.2.6, the sheaf L = 0~ (D) is very ample. The image ~1~1 (Y) is u projectively normal curve and is an intersection of quadrics containing it. We now forrnulate ical curves.

the classical

NoetherEnriquessPetri

theorem

on ca.non-

$2.2. Some General

Results

37

Theorem 2.2.9 (see, for example, Griffith+Harris (1978)). Y c P-l be a smooth canonical CUIW~ of genus 9 > 3. Th,en:

Let

(i) Y i,s projectively normal in P-l; (ii) $9 = 3, th, en _ Y is a plane curve of degree 4; if g > 4, then the graded ideal Iy is generated by components of degree 2 and 3; (iii) the ideal I y is generated by the component of degree 2 in all the cakes except for the following ones: a) Y is a trigonal curve, that is, it has a one-dimension,al linear system gi of degree 3; 11) Y is a curve of genus 6 isomorphk: to a plane czlrue of degree 5; (iv) in th,e exceptional cases a) and b), the Tuadrics passhg through Y c P-’ cut out a surface F c TP9-l wh,ich, is respectkely: a) a nonsingular geometrically ruled surface of degree g - 2 in EJ’gP1 for g > 5, and a quadric (perhaps singular) in P” for g = 4; h) the Veron,ese surface Fd in P”. In the sequel we need to know some properties varieties Y c IPN with the property degY

= codimY

of non-degenerate

projective

+ 1

(such varieties are in some sense extremal since we always have that deg Y 2 codim Y + 1, see Corollary 3.1.4). Let dl > dz > . > d,, > 0 be a set of integers, and let E = Up1 (dl) 8 . @ UP1 (d,,,) be a locally free sheaf of rank m, on P1. We set Y = P(E) (the projectivization of the bundle & on PI), and let T: Y 4 lPL be the natural project,ion. Let C = TT*O~~ (l), and let M = 0 y/p1 (1) be the Grothendieck tautological invertible sheaf (that is, TT,M = &). Let L and All be divisors on Y such that L = 0~ (L) and M = 0~ (A[). Proposition (i)

the

2.2.10.

sheaf

In, the preceding

as generated

M

notakion,

by its sections;

M

we have:

is ample

if and only

if (I, > 0,

V?i=l,...,rn; (ii) (iii)

the

tirnage

h”(M)

PM(Y)

= h0(7r+M)

is a fine

variety;

= h,“(E)

rn = qgI(d,

1);

+

(iv)

the ideal I,, th,e kernel of the natural homomorphism Q: S* H”(M) + $lL>()HO(MW’l), either *is trivial or is generated by elements of degree 2; (v) deg ‘pn/l (Y) = codim Y + 1; (vi) let j be the least index (1 < j < rn) such that d,, = 0; lhen PM is a cone

with

(d,));

P(c~;l:,l0,1 jection

vertex

pl:

P m--J

‘pM(~(@~,7&1 th e restriction x p + pm--J.

))

”

PPJ

~M/~(@):l~JC?~~)

and

base

coincides

isomorphic

with

the

to pro-

Chapter

38

2. Basic Proprrties

of Fano Varieties

Let Y = P(E) b e such as in 2.2.10. Then the variety pm(Y) is called the scrull oJ1’the uector bundle I if it is nonsingular and a cone over the scroll of the vector bundle & if it is singular. Now we come to the classical result we are interested in (SW Fujita (1975), Iskovskikh (1979a), Eisenbud-Harris (1987)). Theorem 2.2.11 (F. Enriques). satisfying th,e condition degY * Then (i) (ii) (iii) (iv) (v) (vi)

= codimY

narzety

+ 1

Y is on,e of the following: u a a a (I a

projective space PN; quadric in, pN; scroll of a vector bundle; cone over a scroll of a vector bundle; Veronese surfuce F4 c p5; cone osver the Veronese surface.

Below

(ii)

Let Y c I? N be a non-degenerate

we describe

some numerical

Proposition

2.2.12.

Pit(Y)

= Z. L + Z. M,

and geometrical

In the preceding

notation,

and KY = -Ernst

properties

of p(E).

we have:

+ (C

d; - 2)L.

r=l

Proposition 2.2.13. Assume in, addition that dl > d,,,. For every integer c, dl > c 2 d,, , we set YC = p(&,)> where I, := @d, +C?pl (d,,), and identify yz with its image in, Y with respect to the embedding determined by the projection, E + E,.. Let u, an,d b be such integers that H’(M@” @ ,C@“) # 0. Then every section, s E H”(M@” 8 f?‘) has zeros of order > q on Y,: if and only if the ineq~uality UC + b + (dl - c)(y - 1) < 0 holds. Proof. For (1, > 0 there exist the following natural isomorphisms: H”(M@:” @ ,C@“) E H”(7r,(M@” @ f.?“)) 2 H”(IFP’, S” I @ Opl(b)), where S” & is the u-th symmetric power of the sheaf E. By assumption, HO(MR” QO CMb) # 0, therefore a > 0. Let ZL = @*1,>&1 (d,); then E = f: cfi E,., and 5’” & = @&S’ E,! @ SiJei I,.. The subvariety I$ c Y is determined by the vanishing of all the sections from H”(p’, E:.) c H”(Y, M). Therefore every section s E H”(p’, S” E 8 c3p1 (b)) h as zeros of order > q 011 x, if and only if s has degree > q in the terms from E:. For this it, is necessary and sufficient that the degrees of all the sheaves of rank 1 occurring in the decomposition of the sheaf S’ &i 8 s”-” &, 8 UPI (b) m t ,o a direct sum be negative for all i < q. This is equivalent, to the condition that the maximum of these degrees (that is. when i = q - 1) he negative. Thus, the desired inequality must hold.

$2.3. Existence

of Good Divisors

in the Fundamental

Linear

System

39

To conclude this section, WC mention one general result which we shall use in Chap. 3 and Chap. 4. This is the multiplication table in the Chow ring of a monoidal t,ransformation of a three-dimensional variety. For the following statement,, see, for example, Griffiths-Harris (1978). Lemma CT:ri’ 4 X

2.2.14. Let X be a smooth complete variety, dim X = 3, and let he the blowup with smooth center C c X. Let 6’(C) = E be the exceptional divisor. Let f he the class o,f u line in th,e ring A(X) 2f E is u plun,e and the class of a fiber Lf E is u ruled surfuc~. The7~ (i) A(X) = n*A(X) ib Z. E @ Z. f as a7~ udd%tave ,9roup; moreover (T,E = a,f = 0, und a,a*A(X) = A(X). (ii) The multipllcut~ve structure in A(z) is determ,ined by the followiny multiplication tuble: a) if C is n point, then

1,)

E”

zz -f,

for

any

if C

E”=-E.f cycle

2

is a smooth

E” = -n*C

x1, E

(C.D)f,

E.a*Z=f.Z=O,

then .f, f

E”

= -

.o*D=O,

deg(&&>

E. f = -1,

VDEA~(X),

WGA2(X).

In add&on,

the us~uul relation deg(Nclx)

where sheuf.

= f . CT*Z = 0

A(X); c~wue,

+ drg(&&

E.o*D=

E. a*2

.9(C)

is the genus

holds: = 2g(C)

of the curve

- 2 - h’x C,

C :

und Nc;lx

is the normul

52.3. Existence of Good Divisors in the Fundamental Linear System Let X be an rL-dimensional Fano variety of index 1’. -Kx N rH, where H E Div(X) 15 ‘Ic f un d amental divisor. If H is very ample, then, by the Bert,ini theorem, a general divisor from the linear system IHI is irreducible and nonsingular. 1%‘~ shall discuss here the question of exist,encr of smooth divisors in IHI when H is ample but not, very ample and the question of existence of divisors with sufficient,ly good singularities on some Q-Fan0 varieties. On a two-dimcnsiona.1 Fano variet,y (that is, on a dcl Pezzo surface), there exists a srnoot,h irreducible curve in the linear system IHI. This is a simple cla,ssical result (see, for example, Iskovskikh (1988)) Demazurc (1980)). A similar result in dirnension 3 wa.s obt,ained by Shokurov (1979a) (SW also Reid (1983b), nlurre (1982)).

Chapter

40

Theorem 2.3.1. Let 1, und let H E Pit(X) IOX(H)I

contains

This

theorem

a smooth

2. Basic

Properties

of Fano

Varieties

be a three-dimensional Fano variety of index r > be a fundamental divisor. Then the linear system,

X

surface.

can now be obtain&

from more

general

results,

namely:

Theorem 2.3.2 (Flljita (1980-84)). Let X be an n-dimension& Fano variety of index r > n - 1, and let H E Div(X) be a f un d amentnl divisor. Th,en the linear system lHI contains un irreducible nonsingular divisor. Theorem 2.3.3 (Reid (1983b)). Let X be a Gorenstein th,ree-dimensional Funo variety with at most cnno,nical singu1aritie.s. Then ( - -k’x( con&ins an irreducible surface with at most Du Val singularities. Now, to derive Theorem 2.3.1 from Theorems 2.3.2 and 2.3.3, it suffices to show that if in Theorem 2.3.3 the va,riety X is nonsingular, and r = 1: then a By Theorem 2.3.3, a genera.1 general surface H E 10~ (H)) 13 ‘. a 1so nonsingular. and therefore /CIx(H)I is not, composit,e surface from 10~ (H)l 1s irreducible, with a pencil since h”(Ux(H)) = h”(0x(-Kx)) 2 4 by the Riemann R.och theorern. If 10x(H) I is base point fret, then the result follows from the Bertini t,heorern. In the opposite case, again by the Bertini theorem, a general surface from I&(H) / can have singularities only in base points of IOx (H) I. Next, it follows from the adjunction formula and the assumption r = 1 that His a h’3 surface with Du Val singularities. Consider a minirnal resolution (T: H + H; then g is a smooth K3 surface. The restrict,ion IHI 1H is a complete linear system, and 11111 := o*IHIIH is also complete. The base locus Bs IMI is nonempty (otherwise Bs lHI 1s also ernpty). According to the well-known result, of Saint-Donat (1974) on linear systems on K3 surfaces, Bs lM = 2 E P’ is a (-2)-curve on fi, and lM = Z+m,lEl, where IEl is an elliptic pencil on l?. Let 2’ be a component of the exceptional divisor for Q. Then Z’.M = 0, hence either Z’n 2 = 8 or 2 = 2’ wit,h m = 2. But, g(Z’) E Bs IHl, wtlence 2 = Z’, m = 2, and Bs IHI is a single isolated point ~(2’) which is an ordinary double of t,his point,, point for a general divisor H. Now let g: 2 --f X be the blow-up and let S be the exceptional divisor. Then l&II = Jo*HJ\,= \H’ + 2SJlg = IN’1 1~ + 22, where H’ is the proper transform of a general divisor from IHI. This contradicts the fact that 2 is a component, of multiplicity 1 in InPI. To prove Theorems 2.3.2 and 2.3.3, we need some techniques developed in the Minimal Model Program by Y. Kawamata and others. We shall reproduce it briefly here following the paper of Reid (1983b). Fix now the following notation and assumptions. (*)

X is an n-dimensional canonical singularities, fundamental Cartier non-empty.

Gorenstein Fano variety of index ‘r with at most -Kx = *rH, where H E Div(X) is an ample divisor, dirn IHI > 0, and the base locus Bs IHI is

s2.3.

Existence

Lemma there

2.3.4

exist

crossings (i)

=

f,E,

the

Y

s,~,ch

f*Kx

Divisors

in the

Fundamental

(1983b), Wilson (1987a)). + X and an eflecttiue

f: Y

Linear

Under divisor

System

the

41

conditions

(*),

C Ei with

nornml

ai

0 implies

that:

+ C

where

ahE,,

ai

E Z,

ai

> 0;

moreover

#

= 0;

f*IHl = r,i > 0,

(iii)

(Reid

II resoktion on

KY

(ii)

of Good

Twhere

ILI+Cr,E,,

und r,, > 0 for

@Curtier

divisor

IL1 is a base point

at least q

f*H

one

free

linear

system,

r,

E Z,

i;

- Cp,

E, is ample

for

som,e

q,p,

E Q,

O 0, t > c + q - r; + A)) = 0 for i > 0, t > c + q - ‘r; + A)) = 0 for c < 1 + r - q.

Proof. Asserkns (i), (ii) f o 11ow frorn t,he Kawamata-Viehweg theorern and the ampleness of N(t), assertion (iii) follows from fact t,hat, E is R fixed component of If* H I. For the following lemma, see Alexeev n = 3 and Wilson (19871~) for n = 4. Lemma

,

(1989),

vanishing (i) and the

and also R.eid (198310)

for

2.3.6.

(i) Ifr>n,-2, thenr, ai = 2 for some i. Then c 5 1-F < l-q. From Lernma 2.3.5 we get that, t,he roots of the polynornialp(t) are -(7~-3), -(n-4), . . , -1 and 1, and that p(O) = 1. The following two cases are possible: of degree 2 n - 2. Therefore 1) f+E = 0. In this case p(t) is a polynomial we have:

Chaptt,r

42

2. Basic: Properties

of Fano Varieties

P(t) = ~ p(t+l)...(t+n-3)(t-I). (71J 3)! But r)(t) = h”(O~(tf*H + A)) > 0 for t >> 0. We get a contradiction. 2) f*E # 0, that is, B := f(E) is a base corr~poru3~tof IHI. In this cast P(t) = (II : l)! (t + 1). . . (t + 12~ 3)(t - 1)(dt - (sn- l)(n ~ 2)) . where cl’ x f*H’L-l On the other

hand,

by the Ricmann~Roc:h

p(t) = x(G’~(tf*H

+ A)) =

&((ff*H+

A)“-’

Compa.ring

E = H”-’

B

t,heortm

E ~ $&h-,

(tf*H

+ A)“-“I,,

+ .

for trrp2T we get

coefficients

(77,- 4)d’ ~ R + 2 = (2AI,

-

ICE).

f*H71-2/fi

If we prove that

(2Aj, this will

- h-E)

f*H”-2jE

give a. contradiction 2Aj,

Wc remark 011e

now that

which

.f*Hr’-‘lE the tlesircd

- KE.

f*H”-21E

> (r ~ 1)d’ = (,uproves

(ii).

Since A > 0, and E > 0, then

= 2A. E. f*H”-’ inequality > (-Kx

3)d’ ,

2 0.

is a consequence - B)B

of the following

. HI’-‘.

(2.3.1)

It is sufficicrd to prove this incqualit,y for the t,wo-dimensional case rest,ricting E ant1 B to general surfaces S E l~rnH1”-’ a.& F t /v~f*H/7zp2 for m > 0. Then S is a normal surface, f: F 4 S is a resolution of singularities, and E and B are curves on F and S rcqectivcly. Decompose f: F + S int,o a composition of 9: F + T and 7r: T + S, where T is a rninimal rc>solution. The curve C = y(E) is Gorenstein, and dtg h-E < tlcg Kc. Next,

Whence Lemma

(leg KF < (B2 + KS) 2.3.6 are proved.

E: that

is. incqualit,y

(2.3.1)

and part

(ii) of

32.3.

Existence

of Good Divisors

in thr Fundamental

Linear

Corollary 2.3.7. IJnder the conditions of Lemma 2:I.G (ii), visor H is reduced and has at most ordinary double singularities sion 1.

System

43

a general diin codimen-

Now we shall prove the existence of good divisors in the fundamental linear system of a Fano variety X (with singula.rities) of index 7’ > 1) - 1. w&c n = dim X. For the following thtorcm: see Reid (1983b) for 71 = 3 and Alcx~v (19X9) for any n,. Theorem 2.3.8. Let X be a cdimensional Gownstein Fan,o oar?;et;y of index I‘ > n - 1 with, at ‘most canonical singularitie,s, and let H be a fundumental Cnrtixr &isor~ o’n X. Thxn the lin,ear system 10~ (H) 1 contains a reduced irreducible divisor with at most canonical singulurities. Proof. Assume that a genera.1 element, from the linear system /H / has worse singularities than canonical ones. Then Bs lHI # 8, and X satisfies conditions (*). By Lemmas 2.3.4: 2.3.6 for t,he corresponding resolution f: Y + X, we have t,hat r,L 5 a,. Therefore INI has no base component,s (otherwise we have u, = 0: T > 1 for the corresponding ET). Next, a general divisor from 1H 1 is nonsingular in codirnension 1 (otherwise there exists I such that ui = 1. 1’ > 2). According to the following result of M. Reid, in such a case a general divisor from the linear system 1H ( 1s an irreducihlc norrnal variety. Lemma 2.3.9 (R.eid (1983b), ,’WC a.lso Fujita. (199Oa)). Let D g Z,,-,(Y) be an effectiw Weil divisor on an n-dimension,al n,orm,al projecti~ur !uar,%et:y Y. AssurrLe th,at D E Div(Y) @ Q D is nef, and D” $ 0. Then the Support Supp D is cownected in codimension 1, thut is, from D = D1+ 02 and D1 > 0, D2 > 0, it follo~~ that dim Yupp ZI, f? Supp Dz > n, ~ 2. Choose a general divisor H from t,hc acljuriction formula7 wc have Kr, = f’*Kll where f’ = f’l,: proved.

L +

1HI a.nd the corresponding

+ c(a,

~ T,)E:,I~,

H; and ui - r, > 0 by Lemma

L from

ILI. By

. 2.3.6. Theorem

2.3.8 is

Proof (of th,eorenr ZFI.?Zj). Choose a general divisor X,,- l from the linear system IHI By Theorem 2.3.8 and t,he adjunction formula.: Xrrpl is a Gorenstein Fano variety of index rl > nl - 2 and dirnension 121 = n - 1 with at most canonical singularities. By t,he Kawamat,a Viehwrg vanishing theorem 1.3.4. the map H”(O,(H)) + HO(Ux,,~l (H)) is surjective. Thcreforc Bs IHl = Bs lHIx,, 1 1. Repeating this process, we obtain a system of cmbedded Gorenstein Fano varictics with at most canonical singularities:

44

Chapter

2.

Basic

Properties

of Fano

Varieties

x,, = x 3 x,, -1 3 . . 3 x2 3 x1 ) dimX, Then

Xi

is

= i,

a smooth

r;>dimXi-2.

curve,

IHI Is,

is

a complete

linear

system,

and BslHI = BsIHI)~~. By the adjunction formula, de IHl Ix1 = H”, Xi E lP1 if T > dim X - 1, and Xi is an elliptic curve if T = dirn X - 1. Thus we get that either Bs IHI = 8 or Xi is an elliptic curve, deg HIxI = H” = 1, and Bs IHl = Bs IHll,, is a simple point,. In the first case, by the Bertini theorem, a general divisor H is smooth. In the second case H is smooth out,side Bs IHI (again by the Bertini theorem). But every divisor H is smooth at, Bs /HI because H’” = 1. Theorem 2.3.2 is proved. Corollary 2.3.10. kt X and H be as in Theorem only if-r = n-l, and $Bs IHI consists of n single point part of X and H.

2.3.8. lying

Then in

Bs IHI # Cn

the nonsingula~

Proof. Choose a general divisor X7&-i from the linear system 1Hl. By Theorem 2.3.8 and the adjunction formula, X,,,- r is an (n - I)-dimensional Gorenstein Fano variety of index T - 1 with at most canonical singularities. By the Kawamata-Viehweg vanishing Theorem 1.3.4, t,he map HO(Ox(H)) + Therefore lHllxvLmI = IH~,7amI 1, and BslHl = H%‘x,,-~ (W) 1s surjective. /. Continuing the process, we obt,ain a system Bs lHl,n-l stein Fano varieties with at most canonical singularities: x,

= x

3 x,-1

. 3 x2 3 x1

3

of embedded

Goren-

)

where dimX, = i, and the index of X; = ri > dimX+ - 2. Then X1 is a smooth curve, IHI Ix, is a complete linear system of degree H’” > 1: and formula Xi = Pi if T > 71,and Xi is it is clear that Bs lHlxl is non-empty

BslHl = BsIH~~,~. By th e acljunction an elliptic curve if T = R - 1. After that only if T = n ~ 1, and H” = 1.

Proof (of Theorem 2.3.3). We use the preceding that the variety X satisfies conditions (*). Lemma 2.3.11. AssurrLe a pencil. Then x(01,) > 2. Proof.

By the Ramanujam

that the linear vanishing

Hi(%(f*H

+ A-L))

By the Bertini theorem, a general divisor the Riemann-Roth theorem for L: . (f*H) h” (UI.(L

s:ystem

We rnay assume

IL1 is not

composite

theorem

(see also 1.3.4),

= 0,

Vi > 0.

L is irreducible

. L + ~(C’L) + c

notation.

(LfE,))

we have

and nonsingular.

= 2 h”(ck(L))

with

2 .9 + 1 )

By

$2.3. Existence where

y = ;H”

zg-2=

f*Hs

+ 1, h”(C?x(H)) > f*H2.L=

The last inequality 1) if 2) if

f (E;) f (ET)

Comparing Ret,urn

of Good Divisors

the preceding to the proof

Linear

System

45

= g + 2. But

f*H.(L+xr,E,).L>

is a consequence

is a point, is a curve,

in the Fundamental

f*H.(L+~&).L.

of the following

remarks:

then f *H . E, . L = 0; then r, > (I,,, (see Reid (1983h)). expressions, of Theorem

we get ~((77~)

> 2.

2.3.3. By the adjunction

formula,

As in the proof of Theorern 2.3.8, we have that either T, > CL,,or f (E,) is a point. Therefore KL = Di ~ D2, where the divisor D1 > 0 is contracted by the morphism f. and 02 > 0. From Lemma 2.3.11 we have p,(L) > 1, and therefore D2 = 0, and the minimal model of L has a trivial canonical divisor. From this it follows t,hat

ai 2 r,

if EinL#O.

(2.3.2)

Applying Lemma 2.3.11 again, we obtain q(L) = 0, that is, the minimal of L is a K3 surface. Apply f* to the formula (ii) from Lemrna 2.3.4: H = S+xrif*Ei

model

,

where

rif&

=

rif(E,)

if E, or if if Ei and

is E, = E,

contracted by the morphisrn n L # Q) f(Ef) is a fixed component n L = 0.

f of IHI

By the connectedness Lemma. 2.3.9 and the assurnption that IL\ is not composite with a pencil, at least one component of the intersection S n SuppCr,f,Ef is a curve. If C c S n f(E2) curve for some .is a movable component f(E,), dim f(Ei) = 2, then, according to (2.3.2), we have T, = 0. If C c S n f (E,) is a fixed curve, then f -’ (C) contains at least one cornponent Ej such that E,i n L # 8. Using (2.3.2) again, we get a.7 > r,i. But in such a ca.se t,he general divisor from 1HI h as multiplicity 1 along C; therefore C @ SUPPC r’tf*E,. This contradiction proves that c ri feEi = 0, S E IHl, and KS = 0. Since for the resolut,ion f 11,: L + S we have KL > 0, then S has at rnost Du Val singular points. Therefore Theorem 2.3.3 is proved for the case when IL1 is not composite with a pencil.

Chapter

46

Lemma ‘with,

2.3.12

a pencil,

OF(F) divisor

then,

= OF, C

Ei

2.

Basic

Properties

(Shokurov

(1979a).

1, =

l)F,

(g +

Reid

where

f * H2 ’ F = 1, and there such

of Fano

Varieties

(19831~)). If IL1 F is a free pen,cil.

ercists

a, unique

is

composite

Isrt, particular,

component

EC) of the

that

f*H.

EO. F = 1: E,

r,f*H.

T(l = 1, a() = 0 ; vi # 0.

F = 0,

Now we change t,he birational model. Let XI be t,hc normalization graph of the birational map X- - +@ determined by the free part linear system IHl. W e may assume that t,here exists a diagram

of the of t,he

Y

Lemma (i)

2.3.13

(Reid

(1983b)).

h’x, = p* Kx, the variety X1 has at most canonical singularities, lix, E Div(Xl), and -iYx is relatively q-ample. 1 - Kx, 1 = (g + 1)lF’I + E’, whe*re E’ is an irreducible surface, IF’1 is a free per&, every F’ E JF’( is a reduced irreducible surface, and F’ n E’ is a redwed irredwible C”U~YJFfor euery F’ E 1F’I.

(ii)

Now very simple the following lemma Lemma surface

2.3.14 which

arithmetic larities,

then

(Reid

h,as

gensus

calculations show (WC use 2.3.7). (1983b)).

a, morphism 1.

h1 (O,l)

q: El

If’ in codimension

that

h1(6~~)

2 9, which

contradicts

Let E’ he an irreducible Cohesr-.-Macaulay redwrd irreducible fibers + IP1 with, on,e E’

has

on,ly

ordin,ary

double

of

s%ngu-

5 1,

Theorem 2.3.15 (Wilson (19871))). Lrt X b e a nonsingular four-dimen- 2H, and let Pie(X) = Z. H. Th,en Fano variety of index 2. Let -Kx the linear system IH1 contakrs aa irreducible n,onkngular divisor.

siona,l

The

proof

Lemma ref2.3.15,

of t,he theorem 2.3.16

a general

(Wilson divisor

is based on the following (1987b)). Un,der the H has at m,ost canonical

lemma. co~nditbns

of

Theorem

singularities.

The idea of the proof of this lemma is based on the technique developed by Y. Kawamata. According to 2.3.7, a general divisor H has at most ordinary double singularities in codimension 1. If S c Sing H is some irreducible surface, then resolving singularities 9: E 4 H, we obtain some double cover $4 s.

52.4.

Base

Points

in the

Fundamental

Linear

System

47

Using the classification of normal surfa.ces due to Sakai, one can prove that, S either is rational or is birationally isomorphic to a ruled surface. From this one gets a contradiction with 2.3.5 (iii). Theorem 2.3.17 (Alexeev (1989)). Let (X, A) be a log Fano variety, and let -(Kx + A) = rH f or some ample Cartier divisor H and positive rational ‘r > dim X - 2. Then the linear system jH1 con I t awns a reduced irreducible diGsor having at most log terminal singularities. Remarks 2.3.18. (i) Flljita’s approach to the proof of Theorem 2.3.2 is based on the notion of Lgenus; concerning this see Chap. 3. (ii) Theorem 2.3.2 is proved in (Reid (198313)) under the weaker hypothesis: divisor -Kx is assumed to be nef and big. (iii) Refer to P rokhorov (1994a), Prokhorov (1995a): Prokhorov (199513)) Prokhorov (1997), Nella (1996)) Mella (1997) for generalizations of these theorems.

$2.4. Base Points

in the Fundamental

Linear

System

According to 2.3.8 and 2.3.11, the fundament,al linear system IHI on a Fa,no variet,y X of index r 2 dimX - 1 with Gorenstein canonical singularities has no base points except for the following case: T = dim -1, deg X = H” = 1. Consider the case T = dimX - 2. Almost everywhere we shall assume that X is nonsingular. The following proposition is an easy consequence of Theorern 2.3.3 and the corresponding statements on K3 surfaces (Saint-Donat (1974)); see Iskovskikh (1979a), Iskovskikh (1988), Shin (1989). Proposition 2.4.1. with at most canonical

Let X be a three-dimensional singularikies. Then

Fano variety

of in,dex r

(i) $r>2, thenBs/-KxI=B; ( as a scheme), and moreover (ii) QdimBsI-KxI > 1, thenBs/-Kx/ -P1 Bs I - Kx I n Sing H = 0 for a general divisor H E I - Kx /; in particular, I-KxInSingX=O; (iii) 2j dirn Bs j - Kx I = 0, then Bs I - Kxl = {point}, a general divisor H E I- Kx / has at this point an ordinary double singularity, and Bs I - Kx I E Sing X. Remark 2.4.2. As well as Theorem 2.3.8, Proposition 2.4.1 is proved in Shin (1989) under weaker hypotheses, namely, -Kx is assumed to be a nef and big Cartier divisor. and X is assumed to have at most canonical singularities. Corollary 2.4.3 (Iskovskikh (1979a)). Let X be a nonsingular threeI - Kxl has dimensional Fano variety of index r. Then the linear system n,o base points ezcept for the following case:

48

Chapter

(**)

2. Basic Properties

of Fano Varieties

r = 1, and on a general surface H E 1 - Kx 1, there exist a curve Z ‘v iP1 and a nonsingular elliptic CILTIJ~ Y such that Y . Z = 1, and 1 - Kx 11H = Z + m,lYI. In, this case Bs / - Kxl = 2.

Lemma 2.4.4

(Iskovskikh

In the case (**)

(1979a)).

from

2.4.:1, we have

3 I 711 < 4. Proof. Let (7: X + X be the blow-up of Z, and let E = o-‘(Z) hc: the exceptional divisor. Then E is a rational geometrically ruled surface IF,!. Denote by F it fiber of IF,, and by C the exceptional section of IF,. The linear systern I - KFI = In*(-Kx) - El h as no base points and contains an irreducible smooth surface E which is a K3 surface. Let ‘p: X + W c P+’ be the morphism determined by the linear system I ~ I(,-I. The fibers of cp are irreducible and reduced elliptic curves and intersect E in one point. It is easy to calculate that W = IV,,, c IP+l is a surface of degree m + 1, and therefore by 2.2.11 W is either a nonsingular ruled surface, or a cone over a rational normal curve, or the Veronese surface. Since W contains a line, the image of a fiber of E = IF,,, t,hen W cannot, be the Veronese surface. We have the following exact sequence for the norrnal sheaf of Z in X: O+

NZ/S

-

NZ/X

+

Ns/x

II QP1C-2)

z

+

0.

II op

(m - 2)

Whence we obtain that Nz,x = &l(a) cfi &l(b), where a + b = rn - 4, a 2 b > -2. If a < 0, then only the following two cases are possible: 1) m = 3, a = 0, b = -1, E z IFI; 2) m=4,a=b=O, E>P1xIP1. To prove the lemma, it is sufficient to prove that the case a 2 1 is irnpossible. Denote C’ := (pPl((p(C)), and F’ := p-‘(p(F)). Then C’ and F’ are irreducible surfa.ces with an elliptic pencil. We have C’ n E = C, a.nd C’ is nonsingular along C; - Kx = C’ + (a + 2) F’. By the adjunction formula, we have -Kc, = (a+ 2)F”, where F” is a fiber of the elliptic pencil on C’. Therefore h”(Oc~(-Kc:~))

> 0,

h>“(0,,)

= 0

From the fact that, C’ has a linear pencil of curves of genus 1, it follows hl(c?~/) 5 1 (see 2.3.14). On t,he ot,her ha.nd; from the exact sequence 0 -0,-(-C’) we get that

h’(Oc;f)

= h’(C?,-(-C’)) (Tlx(KF

From

the exact

-

sequence

0,

-

OCl ----f 0)

= hl(Ox(Kp+C’)).

+ C’) ‘v 0,(-(a,

that

+ 2)F’).

It easy to see that,

g2.4.

we also obtain

Base

Points

in the

< 1, then

System

49

= u+ 1

we desired

the stmctmc

to prove. of Fano

varieties

with

2.4.5 (Iskovskikh (197Ya)). Let X be a nonsingular Fan,o uariet:y of index r, -Kx = TH, H E Pit(X). IHI is base point ,free except for the following cuses:

dimensionnl linear

+ 2)F’))

u < 0, which

The following theorem describes point,s in the linear system IHl. Theorem

Linear

that hl(O,-(-(a

Since h’(C)%)

Fundamental

system

base three-

Th,en

the

(i) r’ = 2, degX = 1, and lHI has a unique simple buse point. In such a case 1H / determines a rutionul map X - - +P2 with elliptic fibers, g(X) = 5, Pit(X) = Z, an,d the mriety X can be realized in one of the following ways:

a) us a do!uble cover ‘p: X + W c p” of u conx over the Veronese swfuce in P5 with a smooth ramification divisor D c W which is cut out on W by u cubic hypersurfuce not passing through the vertex of W; b) us a hyperswrface of de,gree 6 in the weighted projective space Iq1>1,1,2,3). (ii) r’ = 1, g(X) = 3. In such a cuse m = 3 (see 2.4.4), Pit(X) = Z ci) Z, and the variety X is the blow-up of a variety from part (i) ulong u smooth fiber of the rational map X- +P2. (iii) T = 1, g(X) = 4. In such a case m = 4 (see 2.4.4), Pit(X) = ?Z”‘, and the variety X is isomorphk to Fl x p’, where Fl is a de1 Pezzo surfuce of degree 1. Corollary 2.4.6. Let X be a three-dimensional 1. Then the linear system, I- Kx I deternknes where g = i ( -Kx)” + 1 is the genus of X. Frorn

this one can obtain

Corollary

the following

Funo variety u morphism

‘pi -Kx

with I: X

p(X) +

=

iP”+l,

st,atement.

2.4.7. Let X be un n-dimen,sionul Funo variety of index n - 2 that the intersection, of gen,erul fundamental divisors 1. A ssume H1 n H2 n n H,,-2 is a smooth surface. ‘Then the fun,dum,entul kinear system IHl determ,ines a morphism ‘p/fIl: X + ~~+‘L-2, where g = g(X) is th,e genus with

OfX.

p(X)

=

50

Chapter

3. Del

Pezzo

Varieties

and

Fano

Varieties

Chapter 3 and Fano Varieties

Del Pezzo Varieties $3.1. On Some

Preliminary

R,esults

of Large

Index

of Large Index of Fujita

As we already know from Corollary 2.1.13, the index T of a Farm variety X of dimension n does not exceed n + 1. It turns out that the classification of Fano varieties of large index r is rather simple. In particular, if T = II + 1, then X P P’“; and if r = TL, then X E Q c PL+l is a yuadric (see Kobayashi-Ochiai (1973), Kollk (1981), Serpico (1980), and also 3.1.14 below). Let coind X := n+ 1 - T he the coindez of X; then 0 < coind X < sn, and a, natural question arises to cla.ssify Fano varieties with a fixed coindex; the less its value, the simpler is the problem. This approach conforms partly wit,11 the general concept of Fiijita to classify polarized algebraic varieties according to t,heir so-called n-genera. Definition

3.1.1. A polarized varieky is a pair (X, H), where X i,s a *uur%ety, and H is un ample Curtier divisor on X. The de,qree d(X, H) of a polarized variety (X, H) is defined to be the positive integer H”, where n = dim X, the A-genus A(X, H) is defin,ed to be the integesr n + d(X, H) - h”(X, H). projective

Example 3.1.2. Let, X be a Fano variety of index r, and -Kx - rH, where H is a fundamental divisor. Then n(X, H) = 0 for r > n,, n(X, H) = 1 for T = n - 1, and n(X, H) = g - 1 for T = n - 2, where g = $d(X, H) + 1 is the genus of the variet,y X (see Sect. 2.1). Theorem 3.1.3 (Firjita (1975)). A(X, H) > dimBs IHl. In particular, put dimBs lHI = -1). This general

theorem element

Corollary hyperplanx

Let (X, H) be a polarized vwiety. Then A(X, H) > 0 (if Bs lHl = 8, then WP

is proved by induction on dimension of the linear system IHI.

3.1.4. Then

Let X c P”

be un irreducible

deg X > codim Corollary (i) If A(X, (ii) If A(X.

by replacing subvariety

X with not lying

the in, u

X + 1

3.1.5. H) = 0, th,en H is wry ample. H) = 1, th en H has at most isolated

base points.

Definition 3.1.6. A divisor D E lH1 on a polarized variety is culled a rung if it is irreducible and reduced. If the restriction H”(U,(H)) + H”(On(H)) is surjective, then th,e rung is called

(X, H) nap

regular.

ji3.1. On Some Preliminary

Results of Frrjita

51

A sequence o,f varieties X = D,, > D,+l > . . > D1 is mlled dirn D, = i, und every D, is a rmg for (DL+I, H). Definition

3.1.7.

Represent

the

Euler

characteristic

a ladder

tH))

x(X,

in

if

the

fOr.711 dim

X

ck=() I;!

Then the sectional xn--1(X, H). Rem&s (ii) If X = D,, genus of (iii) If

(t

~

&(X,

H)t(t

g(X,

genus

+ 1)

+ k - 1) .

H) is defined

to be the integer

H) = 1

3.1.8. (i) If dirnX = 1, then g(X, H) = h,l(Ox). D is a rung for (X, H), then g(X, H) = g(D, HID), in particular, if > D,-l > . > D1 is a ladder, then g(X, H) = g(D1, HIDE) is the the curve D1. X is nonsinpplar, then 24X,

H) - 2 = (Kx

+ (n + 1)H).

Emrnple 3.1.9. Let X be a Farm variety where H is a fundamental divisor. Then: g(X, where

g(X,

H) =

0 1 i g

of index

HTzP1 . r, and let -Kx

N rH,

if r > dirn X if ‘r = dimX - 1 if r = dim X - 2 ,

g = i H” + 1 is the genus of X (for 7’ = dirn X - 2).

Theorem variety such

3.1.10 (Fujita n = n(X,

th,at

(1977)). Let (X, H) be u nonsingular polarized H) 5 g = g(X, H), and dimBs IHI 5 0. Then,:

(i) (X, H) hus u regular ladder ifd = d(X, H) > 24 - 1; (ii) Bs IHl = f~ if d 2 24; (iii) g(X, H) = A(X, H), und the gruded algebra +c& H”(X, tH) is genernted by elements of degree 1 2j’d > 24 + 1; (iv) all relution,s in @ HO(X, tH) between elem,ents of degree 1 ure genemted f>U by quudric relations 24 + 2. Corollary N rH. divisor. -Kx

3.1.11. Let X be a Fano vuriety of indexsr > dimX-1, an,d let Then the linear system lH1 contains an, irreducible nonsingular

Theorem 3.1.12 (Fujita (1982)). Let (X, H) be a polurized dinrX > 3, d(X, H) > 2, and A(X, H) = 2. Then the linear contains u n,onsingular divisor. that

Corollary genus

g =

nonsingular

3.1.13. 2 or

3,

divisor.

On

th,e linear

a Funo

mriety

system

IHl

X =

I ~

variety system

such

lHJ

of index r = dimX - 2 a,nd +Kx I contains an irreducible

Chapter

52

Now

3. Del

Pezzo

we prove the result

Varieties

and

mentioned

Fano

Varieties

of Large

above on Fano varieties

Theorem 3.1.14 (T. Fujita). Let X he an n-dimensiort,ul of index r > n ~ 1. Then X is one of the following: (i) X = IV, r’ = ‘II + 1; (ii) X = & C P+l is u quadric, r = n; (iii) X = X,, C JPd+7Lp1 is 0, cone ouer a, rational wertez in Ppl, r = 72 - 1 + $; (iv) X = X4 C P’+” is a cone over the Veronese in

pnp3,

7' = rL -

statement,

Corollary 3.1.15. of index r > n. Then (i) X=P, (ii)

X

of large index. log Fano moiety

normal

curve

Ccl C P” with

surface

S4 c P” with

vertex

$.

In all the cases a fundamental For the following Serpico (1980).

Index

divisor

H on X is very ample.

refer to Kobayash-Ochiai

Let X be a (nonsingular) X is one of the following:

(1973), n-dimensional

Kolkr Funo

(1981), variety

r-7-+1;

E & C P+l

is a smooth,

qvadric,

r = n;

Proof (of 3.1.14). F rom the construction described in Example 2.1.5 (vii), it follows t,hat all the varieties listed in (i)- (’iv ) are log Fano varieties of the corresponding index. Conversely, let X be a log Fano variet,y of dimension n a.nd index r > n - 1. Let H be a fundamental divisor on X, and let d = H” be its degree. Using 2.1.14 (i), we get A

=

A(X,H)~n+d-$d(r-n+3)-rr+l=

t,hat is, A(X,

Sjrr-l-r)+l 2, and r’ = (r - (n, - 2))a for some a E W. But r’ 5 3, so only the following cases are possible:

$3.2.

In cases 1) or and X = Q c and HJv, is a of varieties of

Del

Pezzo

Varieties.

1)

r.’ = 3,

2) 3)

r’ = 2, r’ = 3,

Definition

a= 1, a= 1, a = 2,

and

Preliminary

r = n + 1, r = 12, r = 71 + $,

Results

53

d= 1; d= 2; d = 4.

2) we have d = 1 or 2 respectively. Thus, X = ItD” in case l), Y?“+l is a yuadric in case 2). In case 3) we have that X = p2: doubled generator of the group Pic(p2). Using the classification minimal degree 2.2.11, we obtain case (iv).

Remark 3.1.16. T. Sano (1996) classified all Fano varieties dim X - 2 with non-Gorenstein terminal Q-fact,orial singularities. are complete intersections in weighted projective spaces.

of index T > All of them

Further in this cha.pter, we reproduce another general result on Fano varieties. Namely, we describe the classification of Fano varieties of index n. - 1, which are often called in the literature de1 Pezzo varieties.

$3.2. Del Pezzo Varieties.

Definition

and Preliminary

Results

The classification of nonsingular n-dimensional Fano varieties of index n - 1 was obtained by Iskovskikh (1977)) Iskovskikh (1979a), Iskovskikh (1988) for the case TL = 3 and by Flljita (1980-84), Fujita. (1990a) for any 12. Here we consider such varieties as well as their generalizations, de1 Pezzo varieties. Definition 3.2.1. irreducible projective X s,uch that:

A de1 Pezzo variety is u pair (X, H) consisting algebraic variety X nnd an ample Cartier divisor

(i) X has only Gorenstein (not necessarily normal) (ii) -Kx = (n - l)H, where n = dimX; 0 < q < n. (iii) Hq(X,UX(tH)) = 0 Vq,t,

of

un

H on

singulurities;

Remarks 3.2.8. (i) If X is a Gorenstein Fano variety of index n-1 (or, what is the same, of coindex 2) with at rnost ca,nonical singularities, then, by the Kawamata-Viehweg vanishing theorem 1.3.4: condition (iii) of the definition is satisfied, and therefore X is a de1 Pezzo variety. In particular, Fano varieties of coindex 2 are nonsingular de1 Pczzo varieties. (ii) Frorn the Riemarm Roth theorem and condition (iii) of the definition, it follows that /J~~(X, Ox(H)) = H” + n - 1, and n(X, H) = 1. Then, by Theorem 3.1.3, we obtain that dirnBs IHI < 0. For the following proposition, Fujita (1990a): Fujita, (19901)). Proposition 3.2.3. Let X+1 be a generul

Let

refer to Iskovskikh

(1979a),

Flljita

(1980-84))

(X, H) be a de1 Pezzo wwiety, dimX = of the lineur system JHI, and let H,r,-l

element

n. =

54

Chapter

3. Del

Pezzo

Varieties

and

Fano

Varieties

of Large

Index

H~x,,~~. Then (XII-l, H,,-1) is a de1 Pezzo variety (in particular, the divisor XrL-l is reduced and irreducible), und the mup H”(X, O,(H)) 4 HO(X,-l, Ox,,-,(H,,-l)) is surjectkue. From

Proposition

3.2.3 it is easy to deduce

Proposition 3.2.4 (Fujita (1980-84), Let (X, H) be a de1 Pezzo variety, dirnX (i)

Bs 1HI is non-empty single

(ii)

point

which

only

belongs

if

d =

neither

the following

Fujita (1990a), Fujita (199Oh)). = n. and let H” = d. Then:

1, to

such a case Bs IHI con,sists of a Sing X nor to Sing H for u generul

l;n

ti

= 2, th en the linear system IHI determines a double cover PIHI: X 4 if d 2 3, then the linear system IHI determines an embedding y.yjq: x + Jp+y if d > 4, th en the image X = X,1 v P r1+nP2 is an intersection of yundrics.

P;

(iii)

Theorem 3.2.5 (Fujita (1980-84), Fujita (1990a), Fujita (19901,)). (X, H) be a de1 Pezzo variety with dimX = n and H” = d. Then : (i)

2f d = 1, th en X

is 0, h,ypersurfuce of degree 6 in the weighted p,rojective . ,I); if d = 2, then X cun be reulized either as a h,~yper.surfuce of degree 4 in the weighted projective space p(2,1,. . , 1) or us u double cover C+CI: X + IY rumified along a divisor D c IP”” of degree 4; in the last cuse O(H) = cp*(%7L (1); if d = 3, the*n X = X3 c P7L+1 is u cubic hypersurfuce; if d = 4, then X = X4 c IF’n+’ is u complete intersection of two quadrics; if d 2 5, th en the linear s:ystem lHI determines un embedding X = X,i v pr1+‘lp2 with th,e image bein,g un intersection of quudrics. spuce

(ii)

(iii) (iv)

(v)

Let

P(3,2,1,.

Remark 3.2.6. If X is norrnal singular and is not a cone, then d < k(n), where k(2) = 8; k(3) = Ic(4) = 6, k(5) = 5, and k(n) = 4 for n > 6 (see Fujita (1990b)). If one waives the condition that X is normal, then the example from Fujit,a (1990b) shows that d is not bounded.

$3.3. Nonsingular de1 Pezzo Varieties. Statement Main Theorem and Beginning of the Proof We reproduce the classification papers of Fujita (1980-84), Fujita (1979a), Iskovskikh (1988). We remark that in t,he case of class of the divisor H is uniquely

of the

of nonsingular de1 Pezzo va.rieties following (199Oa) and Iskovskikh (1977), Iskovskikh nonsingular determined

de1 Pezzo varieties by the variety X.

(X, H),

the

g3.3.

Nonsingular

Theorem 3.3.1. gree d. Assume thut

de1 Pczzo

Varieties.

Statement

of the

Main

Let (X, H) b e an n-dimensional de1 Pezzo n > 3. Then X l;s one of the followin,,y:

Theorem

variety

55

of de-

(i) d = 1, and X l;s a hypers,ur:face of degree 6 in the weighted projective space q3,2,1;. . 1) (f or other ~realizutions see Isskovskikh (1979a), FQitu (1980-84)). (ii) d = 2, and X is a hypersurfa~ce o,f degree 4 in the wel;gh,ted projectke space p(2.1.. . 1). In this case th,e lin,ear system IHI determines a double couer qq: x ---i P” ramified along a divl;sor D c P” of degree 4. (iii) d = 3, and X = X:3 c p “+I is a cubic hypersurface, U(H) r” C$+I (1). (iv) d = 4, and X = X4 c IV+’ as u complete intersection of two quudrics. (v) d = 5, n 5 6, and X = X5 c IY+” ,’IS u linear section of the Grassmanniun Gr(2,5) c p” (embedded by Pliicker). (vi) d = 6, n < 4: an,d X = Xc; c iYZf4. Th,ere exist three types of s~uch uarietl;es: a) n = 4, and X = j?’ x j?‘” c p8 embedded in iY8 by Segre; 1-j) n = 3, und X = Xfi c p7 is a hyperplune section of a variety of type u) or, what is th,e same, a proje:ctivl;zatl;on, of the tangent bundle of P”;

(vii) (viii)

c) n = 3, and X = p1 x p1 x p1 C p7 embedded in p7 by Segre; d = 7, n = 3: and X = X7 c p” is a blow-up of a point on p3; d = 8, rL = 3, and X = p” with Ox (H) = Or:3 (2).

Remarks 3.3.2. (i) We have that, Pit(X) = Z . H if and only if d < 5. (ii) If d > 5, then the variet,y X = XC1 c ~d+ra-2 is unique up to projective equivalence. (iii) If d > 3. t,hen the va.riet,y X = X,1 c pd+“-’ is projectively normal. Be$nnlng of the proof. From Theorem 3.2.5, we have the description of dcl Pezzo varieties for d < 4 (cases (i) (iv)). Th ere f ore we may assume that d > 5, X = Xd c P’i+rr-2; Ox(H) = O,(l), and X is an intersection of quadrics. A general section X n lY” is a nonsingular de1 Pczzo surface. Thus d < 9 (see Example 2.1.5). Consider now the cascl ‘II = 3 and Pit(X) = Z H. First let d = 9. Then a general surface S E /EIl 1s a de1 Pezzo surface of degree 9, that is, S Y p”, and Us(H) = C&Z (3). Th e restriction map Pit(X) + Pit(S) = Z is injective, and its cokernel is torsion-free (Iskovskikh (1979a)). From this it follows that, there exists an clement, L E Pit(X) restricting to 0~2 (1). Further, H = 3L, and -Kx = 6L. This cont,radicts r = 2. Similarly, we exclude the case d = 8 and n = 3 if a general divisor S E IHI is isomorphic to l?l x P’. Therefore we can assurrle from now on that 5 < d 5 8, and that the general hyperplane section S E IHI is a de1 Pezzo surface of index 1 and degree d; in particular, X contains lines. Now we describe some results on families of lines on de1 Pezzo varieties (they will be used in Sect. 3.4).

56

Chapter

3. Del

Pezzo

Varieties

and

Fano

Varieties

of Large

Index

Definition 3.3.3. By a line (conic) on a polarized variety (X, H) we shall a reduced irreducible curve Z c X such that Z . H = 1 (respectively, Z.H=2), andZ=p’. m,ean

As was mentioned above, if (X, (X, H) # (p3, 0p:{ (2)), then there 3, then lines on X are usual lines Xd of iFdfTap2 determined by the

H) is a nonsingular de1 Pczzo variety, and exists a line on X. Moreover, if d = deg X > under the projective embedding 91~1: X = linear system JHJ.

Lemma 3.3.4 (Iskovskikh (1979a)). Let (X, H) be a nonsingukr th,reedimensional de1 Pezzo variety of degree d, (X, H) # (p’, C+(2)), and let Z c X be a line. Then there are only the following possibilities for the normal deaf

of Z

in X:

(0,O)

: Nzjx

-1) (2, -2) (3, -3)

: Nzlx : (only

(1,

= C&I

: (only

(3 Up1 ;

= 0,1(l) @ opl(-l); ifd = 1 or d = 2) NZlx if d = 1) Nz/x = Opl(3)

= Up1 (2) CDU,I (-2); @ 0p1(-3).

Denote now by r = T(X) the Hilbert, scheme parametrizing and by S = S(X) the universal family of lines on X. Consider diagram:

lines on X the natural

I’

s-x

r Proposition (i)

3.3.5

(bkovskikh

If d 2 3, th e scheme

r

(1979a)).

= r(X)

projective

is smooth,,

and of pure

dimen-

sion 2. (ii)

Let d > 3, and let T,: Si --) l7~ be a universal family over a component ri c r. Then only the following two cases are possible: a) pi: S, --f X is a surjectiue map; in this case a general line Z from the family

b) pi(S,) from

EL is of type =

P2

the family

(0,O);

is a plane ri

is

on

X

=

X,f

c

P”+l;

in

this

cuse

every

line

of type (1, -1).

Rem,arks 3.3.6. (i) I n case b) from 3.3.5 (ii), we have Pit(X) # z. H, and p(X) >_ 2. Let p(S;,) = E = p2. Then for any line Z c E c X, we get from the exact sequence 0 ----) Nz/F:

+

Nz/x

-

NE,x]~

--f

o

that Z. E = cl(N z/x) - cl(NZ,E) = -1. In other words, the divisor E can be contracted into a. nonsingular point. (ii) In the case CE= 2, the surface r can have singular points (corresponding to (2, -2)-lines), see Tikhornirov (1980a). If d = 1, then the surface r is not reduced Tikhomirov (1981).

53.4.

Del

Pezzo

53.4. Del Pezzo Continuation We continue to proof (1979a), Iskovskikh (1988).

Varieties

with

Picard

Number

p = 1

57

Varieties with Picard Number p = 1. of the Proof of the Main Theorem Theorem 3.3.1, following In th’ is section we consider

papers by Iskovskikh the case p = 1.

b e a nonsingular Proposition 3.4.1. Let (X,H) three-dimensional de1 Pezzo variety of degree d > 3 with Pit(X) = Z H. Let Z c II be a line, and let a:X + X be the blow-up of 2. Let H* = a*(H), and let E = a-‘(Z) be the exceptional divisor. Th,en, X is a Fano variety with p(X) = 2. Therefore there exist two extrernal rays R, R’ on, X (see 1.3.8). Moreover, if o: X -+ X is the contraction of R, then there exist only the following possibilities for the contraction +!J of R’: (i)

(ii) (iii)

(Cl)-type, d = 3, and ‘p: X- + p” is a conic bundle with the degeneration curve C c IP2 of degree 5. The curve C is smooth if Z is of type (O,O), an,d has a single ordinary double point if Z is of type (1, -1). El-type, d = 4, and p: X + p” is a blow-up of a sm,ooth curve of degree 5 a,nd gen,us 2. The exceptional divisor D is equivalent to 2H* - 3E. El-type, d = 5, and ‘p: X --j Q is a blow-up of a smooth curve C of degree 3 and genus 0 lying on a nons~agular yuadric Q c p*. The exceptionol divisor D is equivalent to H* - 2E. In th,is case C lies on a smooth hyperplane section if Z is of type (0,O) and on a singular hyperplane section, if Z is of type (1, -1).

Sketch of the proof. The 1inea.r system IH* -El on X is base point free, and -I‘+=2H*-E=H*+(H*-E).H ence X is a Fano variety. Let ‘p: X --i Y be a contraction of an extrernal ray different from c. Consider for example the case d > 5. From the Riemann-Roth theorem and the Kawamata-Viehweg vanishing theorem, we get dim jH* -2EI = d-5. Therefore if d > 6, then there exist, irreducible divisors D1 N H* - 1;i E and D2 - H* - kzE, kl , k2 > 2, and DI # D2. But an easy calculation shows that 0 < D1 D2. (H* -E) = d - kl ks - kiIc2, that is, d > 8, and dim IH’ - 2El > 3. Choosing 03 N H* - k: 2, and 0:~ # D1, 03 # Dz, we get 0 < DI 02 Ds = d - kik2 ~ kilc:~ - k2k3, that is, d > 12 which contradicts d < 3 (see Sect. 4.2). Consider now the case d = 5. There exists a unique irreducible divisor D N H*-kE, k > 2. Since0 < D.(H*-E)2 = d-2k-1, then D N H*-2E, and the linear syst,em )H* ~ El contracts the divisor D. Therefore dirn Y = 3, by the linear systern a.nd the contraction rnorphism ‘p: X ---f Y is determined In(H* - E)I, n > 0. Since (H* - E) H* D = d ~ 2 > 0, then y(D) is not a point. Hence by the classification of extremal rays (see 1.4.3), ‘p is of type El, the variety Y is nonsingular, and p(D) is a smooth curve. Further, Tl icrefore Y is a Fano variety of index 3, @-KY) = 4% + D = 3(H* -E). that is, Y = Q c p4 is a qua.dric (see 3.1.15), and one can take n = 1. Part (iii) immediately follows from a.11 of this.

Chapter

58

3. Del

Pezzo

Varieties

and

Fano

Varieties

Index

of Large

Corollary 3.4.2 (Iskovskikh (1979a)). A n~onsinyular three-dl;rrben,sior~~~l de1 Mezzo variety (x, H) of degree 5 is uniyue up to isornorphism and can be realized as a smooth section of the Grassrnannian Gr(2, 5) c p” (under the Pliicker embedding) by a subspuce of codimension 3; Pit(X) = Z H. Proof. From the adjunction formula and the Lefschctz theorem on hyperplane sections, one can easily deduce that a smooth section Gr(2, 5) n p” satisfies the hypothesis of 3.4.2. From 3.3.4 and 3.3.5, it follows that X contains a line of type (0; 0). Then, using the previous notation, C is contained in a nonsingular hyperplane se&ion Q. The group Aut( Q) acts transitively on the set of sue% curves. From this we get that X is unique. Now let (X, H) be a four-dimensional nonsingular de1 Pezzo va,riety wit,11 Pit(X) = Z H of degree d. From the Lefschetz theorem on hyptrplane sections and Proposition 3.4.1, it follows that d < 5. Since t,hr casts d < 4 a.re considered in 3.2.5, we consider here t’he ca.se d = 5, that is, X = X5 c p7. Lemma

3.4.3

(Fujit,a.

(1980-84)).

There

exists a plane

on X.

Proof. It is clear that X contains a lint 2. As in Proposit,ion 3.4.1; let CT:X + X be t,he blow-up of 2, let. H* = o*H, and let E = ~‘(2) be the except,ional divisor. Then 2 is a Fano variety, and the linear system /H* - El deterrnines t,he cont,raction of an extrcmal ray ‘p: 2 + Q c p’. It is easy to calculate that (H* - E)4 = 2, that, is, Q C p” is a follr-dimensiorlal yuadric. If X does not contain planes, then cp cannot have fibers of dimension 2 2. of a smooth surfac*c: According to Theorem 1.4.5, 9: 2 4 Q is the blow-up F on the smooth variety Q. L_et, Y c X he a nonsingular hyperplane section passing through 2. Let Y c X he its proper transform, and Q’ = p(Y). From Proposition 3.4.1 it follows that, Q’ c Q is a. nonsingular hyperplane section. and F n Q’ = C is a smooth curve of genus 0 and degree 3. Whcncc WC get, that F = F:s c IID is a surfa.ce of degree 3, and F y FI. The proper transform a(y(L)) of the (-l)- curve L on F = F1 is t,hen a plane on X. The

following

lemma

can be proved

in a similar

way.

Lemma 3.4.4 (Fujita (1980-84)). Let X = X5 c p7, and let P c X he Q pla,ne. Let CT:2 + X be the blowup of P. Let H* = 8H, and let E = aP1(P) be the exceptional divisor. Then the linear system lH* ~ E/ determines a morphism

and morphism (i) (ii)

+

P4.

One

has

only

the

following

possibilities

for

the plane

P

‘p:

three-d~rnen,sion,al quadric, and u general = Q c p4 is a smooth of qx 2 + Q is isomorphic to IID’ (I’ is a cT:l.l-plane); +T 2 + Pi is the blow-up of a smooth rational normal CUT~I(: C = C3 c P” C IP4 (P is a cTz.n-plane).

~(2) fiber

Lemma a,lso

qx 2

condains

3.4.5

(Fujita

a a2,2-plane.

(1980-84)).

If th,ere

is

a a:~,1-plane

on

X,

then

X

$3.4.

Del

Pezzo

Varieties

with

Picard

i+oo~. In the opposite case in the situation of cp: X + Q has no two-dimensio@ fibers. Then it is a l?‘-bundle. Therefore xtop(X) = axtop = xtup(X) = 5. If Y C X is a smooth hyperplane for i 5 2, and 0 = bs(Y) > O:,(X). H ence hd(X) where 7 E H2(X, Z) is a primitive element. By which contradicts the equality 5 = [HI” = k’y”.

Nurnbrr

3.4.6

(Fujita

59

Lemma 5.4.4 (i), we gzt that is easy to see tl1a.t cp: X + Q 8. From this we obt,ain t,hat section, then h+(Y) = b,(X) = 1. But then [HI” = ky, Poincark dualit,y, y2 = ztl,

Thus, we have proved that there exists a a2.2-plane const,ruction 3.4.4 (ii) can be carried out. The group Aut(p”) acts transitively on the set (7% c p’ c p”; therefore X = X5 c P7 is unique up to deduce frorn the adjmlction formula that X can Gr(2,5) (similarly to 3.4.2). The case rl = dirnX > 5 can be considered in a the following birational correspondences between X Proposition

p = 1

on X = X:, c p7. so the of ra.tional normal curves to isomorphisrn. It is easy be realized as a s&ion of similar way. One uses here and p” (Fujita (1980-84)).

(1980-84)).

(i) Let X = X5 c P8 he a five-dimensionml de1 Pezzo mriety o,f de,qree 5 with Pit(X) = Z.H. Th en X con~tains a plane P, and there exists the following diagram associated to P:

(ii)

wh,ere g: 2 + X is the blow-up of P, and p: 2 + I?” is the blow-up of a smooth surface F = F:s c p4 c IP’” of de,yree 3, F e Fl. The morphism 9: 2 + p” is determined by the linear system lH* ~ El, where H* = cp”H, und E = 0-l (I’) is th,r: ezceptionul disvisor of CT:2 ---f X. Let X = Xj c Pg be a six-dimens*ional de1 Pezzo mriekg of degree 5 u~ith Pit(X) = Z.H. Then X contaks u plane P, an,d th,ere rzists the followCxg diagram associated to P:

where c: 2 + X is the blow-up of P, and ‘p: 2 --j p” is th,e blow-up of ~1,smooth vuriety V = Vs c p5 c p” of de,gree 3, V 2 p1 x P2 embedded by Segre. Th,e morphism qx 2 + p” is determined by the linear system IH* - El (H* and E ure the same as in (i)).

Chapter

3. Del Pezzo Varieties

and Fano Varieties

of Large Index

In a way similar to that of 3.4.2, one can deduce that, the variety X is unique up to isomorphism using Proposition 3.4.6 (i), (ii) and the representat,ion of such varieties as sections of the Grassmannian Gr(2,5). There are no de1 Pezzo varieties of degree 5, dimension > 7 and Pit(X) = Z H. This follows from t,he general fact: Theorem 3.4.7 (Fujita (1981). Fujita (1990a.)). The Grassmnnm Gr(r,n) cannot be embedded as un ample divisor into n smooth all the cases, excluding r = 1, r = n - 1 and (r, n) = (2,4).

variety variety in,

A three-dimensional de1 Pezzo variety X = X5 possesses many remarkable properties. For example, a.s Mukai-Umemura (1983) have shown, it is a quasihomogeneous space with respect to the action of the group SLz(@). Denote by iW(j the space of binary forms of degree d in variables to, tl with the natural action of SLp(c). It is well known that every irreducible representation of SL2 (@) is. isomorphic to some h/l,. Theorem 3.4.8 (Mukai-Urnernura (1983)). Th,e variety X5 c I?” cm be realized as the closure of the SLz(@I) - or b’t z of some point [x6] E p(MG), where of the binury octahedral group 0 c xfi = tCltl(t;f + tf) is a semi-invariunt SLz(@) of order 48. Proof. The group SLz(c) acts on t,he Grassmannian G = Gr(2,5) = Gr(2, b14) c IF”(~~n14). By the Clebsch-Gordan forrnula, r\“bld F nrl, @ h1~. Consider the invariant section X = G n l?(&&j) = G n p”. If dim X > 4 or if X is reducible, then there exists a component X1 c X such that deg X1 < 2 a.nd dim X1 = 3 or 4. But in such a case X1 is contained in an SLp(@)-invariant subspace p” C p(nfe), k < 5, which contradicts the irreducibility of n/16. Thus x = xr, c P(M~) is an irreducible SLz(@)-’ mvariant (perhaps singular) de1 Pezzo variety. Considering a general curve-section, it is quite easy t,o obtain that either X is normal or Sing X is a linear subspace P2 c X c p(IL&). The second case cont,radicts the irreducibilit,y of A&,,. Thus WC get that Sing X is an SLs(@)-invariant, curve B c X. But the space p(&) contains only one SLz(@)-invariant curve, a rational normal curve B = &j C p(hf(j) = p” (see Mukai-Urnernura (1983)). Let H be a general hyperplanc section of X. By the adjunction formula, H is a Gorenstein normal de1 Pezzo surface of degree 5. According to the classificat,ion of such surfaces (see, for example, 2.1.5 (ii)), a minimal resolution fi + H is isomorphic to the blow-up of 4 points on Y2, in particular, p(H) = 5. But H is singular at 6 points H n B, and so p(H) 2 7. This contradiction shows t,hat X = X5 c p” is a nonsingular SL2 (c)-’ mvariant de1 Pezzo variety. Further, the octahedral group 0 contains a subgroup of index 2, t,he bina.ry tetrahedral group lI’, which has two semi-invariants x4, x& E 1114 of degree 4. The element ~1 A xi E r\2n& is a semi-invariant, of 0, so xq~x& = 26: E MC; c //“iW4. Therefore SL2(@)[z(;] = X. Remark 3.4.9 (Mukai- Umemura contains the following SLz(@)-orbit,s:

(1983)).

The

variety

X

= X5

c

p(i&)

s3.4. Del Pezzo Varieties

with Picard

Number

p = 1

a) the open orbit SLz(@) . [~(i] isomorphic to SLz(@)/O; b) t,he two-dimensional orbit SLz(@) . [tot:]; c) the one-dimensional closed orbit B = SLz(@) [ty], which normal curve of degree 6 in l?‘.

61

is a rational

The union of orbits SLz(C) [tasty] U SLz (C) . [t:] 1s an irreducible surface S c X, which is cut out on X by an SLz(@)- invariant) quadric. The normalization of S is isomorphic to p1 x Il’)l with the diagonal action of SL2(@), and the map v: ILD’ x p’ 4 S is determined by some (non-complete) 1inea.r system of bidegrec (1,5). THe surface S is singular along B. In the next section we shall discuss other examples of t,hree-dirnensiona.1 Fano varieties which can be represented as quasi-homogeneous spaces. The variety X = X:, c p” also has an interesting relation to the problem of cornpactifications of C” (see Hirzebruch (1962)). For the following theorem, refer to Pet,ernell-Schneider (1988), Furushima (1986): Furushima-Nakayama (198913). F urushima-Nakayarna (1989,). Theorem 3.4.10. Let X be a nonsingular three-dimensl;on,al prqjective variety, an,d let D c X be an irreducible divisor such that there exists o, biholomorphic isomorphism X \ D + C”. Then X is a Fano variety with Pit(X) = i3,. D. Moreover, if -Kx = rD (that is, r i,s the index of X), then (i) r >4+ (X,D) = (p”,p”); (ii) r = 3 * (X, D) = (Q, Q’), &err Q c p4 l;s a smooth quadrk, and Q’ is its singular hyperplane section; (iii) r = 2 ===+ X = X5 c I?)“, and one has onSly the follo~w%n,g posslbilitks ,for the divisor D : a) D is a n,orm,al Gorenstein de1 Pezzo swjace ,w%th a sk~gle Du Vu1 singular point of type A4; b) D is a hyperplane section singular along a (1, -l)-line (see 3.3.4). To prove the existence of compactifications a) and b), one uses the COIIstruction from 3.4.1 (iii). The case r = 1 will be discussed in the next, section. Remarks 3.4.11. (i) According to K. Kodaira, every projective compactification X of the space @” with an irreducible divisor D (where X \ D rv Cc) is a Fano variety with p = 1, and D is a fundamental divisor on X. From 3.1.15 it follows that if the index r of X satisfies r > sn, then for (X, D) only the trivial cases are possible. (ii) The second aut,hor has studied compactifications of C4 of index 3 (Prokhorov (199413)). He proved that if (X, D) is such a compactification, then X = X5 c p7 is a de1 Pezzo variety of degree 5, and there are exactly four possibilities for the divisor D.

62

Chapter

3. Del

Pezzo

Varieties

and

Fano

Varieties

of Large

Index

53.5. Del Pezzo Varieties with Picard Number p > 2. Conclusion of the Proof of the Main Theorem FirstL> let’ X = X,1 c l?“+’ be a nonsingular three-dimensional de1 Pezzo variety with p > 2, and -Kx = 2H. According to 3.2.5, we can assume that d > 5. The Mori cone NE(X) is genera.ted by extrernal rays RI = Since -Kx R,[lll,. . . , R,,, = Iw+ [/7rL]. Let yi: X ---j Y, be their contractions. is divisible by 2 in Pit(X), then p(Ri) = 2, and the rays RI,. , R,,, can be of type E2, C2; or 02 (see 1.4.3). Proposition 3.5.1. Let (X, H) be a nonsingular p(X) > 2. Then on,e h,us th,e following possibilities extremal rays pL: X + Yz, i = 1,. . , m:

de1 Pezzo wuviety with for th,e con,tro,ction,s of

(i) deg X = 7, und ~1: X + IP’ is of trrpe E2, and 92: X --+ p” is of type 6’2, m = 2; (ii) degX = 6, and (pI: X --f IP2 ure of t:y;ue C2, i = 1,2; (iii) degX = 5, and y~i: X + P x P are of type C2, i = 1,2,3. Lemma 3.5.2. For cver?y pod of X, theTe is a, he IFI such that C . H = 1) puwing throw& it.

(th,ut is, u curue C !x

Proof. Since p(R,) = 1, then li. H = 1, and if one of the rays RI.. : R,,, is of type C2 or 02, then we can take for C the corresponding represcntativc I,. Assurne that all the R+ are of type E2. Choose a curve-section r E H2; then r z c ail,, (Y; > 0. But this implies that r Ei < 0: where E; is the exceptional divisor for 9,. This contradicts the ampleness of H. Assume

that

one of the rays Xi is of t,ype E2.

Lemma 3.5.3 (see Tori-Mukai (1986), W’ lbniewski (1991b)). Let (T: V + W be the blow-up of a point on a nonsingular variety W, and let V be a Fun,o variety. Th,en, W is ulso a, Funo vari&y. Proof. If W is not a Fano variet,y, then --(TYKE R < 0 for some extremal ray R on W which is not contracted by the morphism 0. But in such a cast the exceptional divisor E of morphism 0 has negative intersection with R, and therefore the exceptional locus of t,he ray R is contained in E which is a fiber of g. This is impossible by t,he definit,ion of extremal rays. Frorn Lemma 3.5.3! it follows tl1a.t U, is a Fano variety of index sr = 2 or ‘r = 4. If T’ = 4, then Y, e p”, and we get the case 3.2.1 (vii). If Y, is of index 2. then (-KY,)’ = (-I(x)’ + 8, that is, deg Y; = dcgX + 1 > 5. It follows from the results of 3.4.1 that p(YL) > 2. By Lernrna 3.5.2 there is a line passing through the point cp( EL). Th en its proper tra.nsform on X has zero intersection with -1(x. This contradicts the ampleness of -Kx. Therefore we rnay assume that all thr rays Rl, , R,,, are of type C2 or 02. Consider first the case p = 2. Then 0lere are only two extrernal rays RI, R, on X. and Yi = p1 in case 02 and k; = p” in case C2.

53.5.

Del

Pezzo

Varieties

with

the

Picard

Number

p > 2

63

Lemma 3.5.4 (Tori-Mukai (1983a), see also Chap. 7). 1fp(X) = 2, then Pit(X) = q$ Pic(Yr) @(pa Pic(Y,). Moreover -Kx = 2H = 2H1+2Hs, where H+ is a non-negative generator of (pb Pic(Y,). The morphism ‘p = yr x ~2: X + Yr x Yx is finite by the definition of extremal rays. Therefore if RI is of type D2, then R2 can only be of type C2 and we have a finite surjective morphism w: X + P1 x p2. It is easy to calculate deg p = HI Hi = HI 12 = i(-Kx) 12 = 1, that is, X = p’ x p2. But we should have in this case that r(X) = 1. Now let X1 and R2 be of type C2. In such a case ‘p: X --f p2 x p2 is a finite surjective map of X onto a divisor X’ c l?’ x p”. Sirnilarly to the preceding case, we get that X’ c p2 x p2 is a divisor of bidegree (1,l): and that cp: X + X’ is an isomorphism. The assert,ion 3.3.1 (vi) a) is proved. Consider finally the case p > 3. Then all the extremal rays RI, . . . , R,,, on X are of type C2, m > p(X) > 3. Since H . li = 1, and a general H is a nonsingular de1 Pezzo surface, then pill: H + Y, is a birat,ional morphism, and Y, is also a de1 Pezzo surface with p(Yi) = p(X) - 1 > 2. If p(X) > 4, then X E Y, x p’ (see 7.1.15), and therefore X cannot have index 2. In t,he case p(X) = 3, the variety X is a Fano variety of index 2 if and only if Y; E pr x p’, and X z pr x Y, ‘v p’ x p’ x p1 (see 7.1.29). Consider now the case dimX > 4, p(X) > 2. Lemma 3.5.5 (WiSniewski (1990a)). Let X -Kx = rH, und dimX > 4. A ssume r 2 2, contains u nonsinqular divisor. Then

NE(X)

be a Fano anriety o,f index that the linear system 1HI

= NE(H)

Proof. By the Lefschetz theorem on hyperplane sections, p(X) = p(H). It is sufficient to prove that every contra.ction of a,n extrernal ray ‘p: X + W contracts some curve lying on H. This is true if cp has a fiber of dimension > 2. If cp has no fibers of dirnension 2 2, then, according t,o Theorem 1.4.5, cp: X + W is a standard conic bundle (case (i) from 1.4.5 is impossible because T > 2). Since a general fiber of y is a smooth conic, then -Kx.C = rH.C = 2. Thus H C = 1, and T = 2, that is, p: X + W is a pr-bundle, and H is its section. Further, p(H) = p(X) = p(W) + 1, so the birational morphism 91~: H ---f W has a connected fiber of positive dimension. From 3.5.1 it follows that degX can be equal only to 6 or 7. Consider for example the case degX = 6, p(X) = 2. Then there are two extremal rays RI, R, on X. Let ‘p7: X ---f Y, be t,heir contractions. By Lemma 3.5.5, p71H is a contra,ction of an extremal ray on a smooth 1rypqAane section H c X = X7 c pg. Frorn 3.5.1 (ii) it, follows t,hat p21~: H + P” are @-bundles. Therefore the fibers of p,: X --i Y,, are two-dimensional, and Y, y p2. We get a finite surjert,ive morphisrn

Chapter

64

3. Del Pezzo Varieties

and Fano Varieties

of Large Index

But, in such a case Kx = p*Kpxp + B, where B is the ramification divisor of cp. The only possible case is B = 0, hence 9: X + IID” x p2 is an isomorphism. The case degX = 7, p(X) > 3 as well as the cast dimX > 5 can be considered in a sirnilar way (see Fujita (1980-84)). The proof of Theorem 3.3.1 is completed. Using classification of Theorem 3.3.1, one can obtain a rnore deta.iled description of families of lines on thrt:c>-dirnensional de1 Pezxo varieties. Proposition 3.5.6 (Iskovskikh (1979a)). Let (X.H) # (pi:Up:~(2)) be a three-dimerLsiorLa1 non.sing~ular de1 Pezzo uarkty of degree d. Let r = r(X) be the base of the fnmdy of &es on X. Then: (i) Zf d = 7, then th,e sch,em,e T is a disjoint union (ii) if d = 6 and in case 3.3.1 (vi) b), th, e s&me r copies

(iii)

if

(iv)

copies

is u disjod

of JP2; u*wion

of two

of P2;

d = 6 and in

three

of two

cop&

(see also

cuse 3.3.1

of P

(ui)

c), the scheme

r

is a disjoint

union

of

X IP;

Furushirnu-Nukul/umu

(1989a))

if

d = 5, then r

is isomorphic

to P.

Remark 3.5.7 (FurushimaNukuyamu (1989a), Prokhorou (199Oc)). From 3.4.X and 3.4.9 it, follows that for d = 5 there is a natural action of SL2(@) on r E p2. In this case the family r decomposes into t,wo SL2 (Cc) orbits, namely: a) a closed orbit r, c r which is a conic on r = p2; lines from this subfamily are of type (1, -1) and sweep out a divisor S N -Kx (see 3.4.9); 1)) an open orbit fil = r \ r,; lines frorn this subfamily are of type (0; 0). The urliversal farnily of lines 7r: E + r E p2 is the projectivization E = ~Pz (E’) of a vector bundle E’ = t,C on p2, where t: p1 x lY1 + P2 is a double cover obtained by taking the symmetric square, and L: = &I xpl (0.3). The morphism p: & - X = X5 c p’ is a finite morphism of degree 3 rarnified along the divisor x-‘(rl) = p-l(S). To the conic rl c p” = r, there corresponds a non-degenerate symmetric bilinear form q(., .) on p2 = r. The condition that two lines lying on X = Xc, c p” intersect can be easily expressed in terms of t,his form, namely: two distinct lines Cl; Cz on X int,ersect if and only if we have q(cl, ~2) = 0 for the corresponding points ~1: Q E r. For a line C c X and the corresponding point c E r, the condition q(c, c) = 0 holds if and only if the line C is of type (1, -1). Proposition

3.5.8.

(i) Let X = X4 C IP5 be a nons*ingulur intersection of two quadrics. on X is an ubelian surface base of the family of lines 1’ = r(X) th,e Jucobiun of u CILTIE of genus 2 (see Gr-ifiths-Harris (1978)).

Then

the

which

is

$4.1. Elementary (ii)

Rational

Maps: Preliminary

Results

65

Let X = X:3 c IP4 be a nonsingular cubic hypersurfuce. Th,en th,e base of the family of lines r = T(X) is a nonsingular surface with the following invarinnds: q = 5, pg = 5, xtop = 27 (see Tyurin (1970), Tyurin (1971), Altman-Kleiman (1977), Bom,bierikSwinnerton-Dyer (1967)).

Remark 3.5.9. Families of linear subspa,ces on cubic hypersurfaces X = x3 c lP+l and on intersections of two quadrics X = X4 c IlJ’&+’ are studied for arbitrary n in Altma,n-Kleiman (1977) and M. Reid’s t,hesis. Linear subspaces on de1 Pezzo varieties X = X:, c IYf3 of degree 5 (sections of Gr(2:5) c p”) are studied in Frijita (1980-84). Refer to Tikhomirov (1981), Tikhomirov (198(&r), Welters (1981) for families of lines on de1 Pczzo va,rieties of degree 1 and 2 and dimension 3. Not,e that in the cases d = 1,2 the surfa.ce T(X) can be singular, and for d = 1 it can even be non-reduced along a curve.

Chapter 4 Fano Threefolds with 54.1. Elementary

R,ational

Nfsps:

p= 1

Preliminary

R,esults

In t,his section WC generalize constructions considered by Iskovskikh (1978), see also (Iskovskikh (1989b), Takeuchi (1989), Cutkosky (1989)). Each such construction is a particular case of Sarkisov’s elementa.ry link (see Sect. 9.2). We state the result,s only for the case of charact,eristic zero, for the ca,se of positive characteristic see (Shepherd-Barron (1997)). Let X be a Fano threefold of index r and let H be a furldamemal divisor on X, i. e. a divisor such that -K x = rH with maximal r. Let C c X be a smooth curve (or z E X be a point). Denote by O: X + X the blow-up of X with center C (respectively, with center z). We assume that C (respectively, r) satisfies the following two conditions: (*)

the anticanonical divisor -“x is numerically (-Kg)” > 0 (i. e. -I(,is big), (**) there are no effective divisors D on X such that Lemma tem

4.1.1.

/-nKx/

Under

f or some

assumptions n > 1 determines

effective (-KF)’

(nef)

and

. D = 0.

(*) and (**), the complete linear sysa birutional morphism 9’: i? + X’ C

IP” such, that (i) the varieky X’ is normal, Q-Gorenstein luritjes, (ii) ‘p’: X + X’ is a small resolution (i. e. one),

ur~cl

it is un

has at worst

terminal

sin,gu-

isomorphism

in codimension

66

Chapter

(iii)

X’

is nonsingular

4. Fano

if and

only

if and only if 2 is a, Funo

Threefolds

with

p = 1

if ‘p’ is an isomorphism,

threefold

with

p(g)

which

in

turn

holds

= 2.

The existence of ip’ follows from the Base Point Free Theorem 1.3.6. It is easy to see from (**) that cp’ does not contract divisors, i. e. y’ is a smnll contraction, and singula.rities of X’ are terminal. Finally, (iii) follows from the fact t,hat p’ is determined by a mult,iple a.ntica,nonical linear system. According to Theorem 1.4.15 and by Lemma 4.1.1, there exists a flop x = ((~+)-~ocp’:%--+x+> with

Therefore

the divisor

(see 1.3.7). then we put and 1.3.13, such a way We have

-Kz+

is nef, and there exists

an extremal

ray R on x+

+ Y be its contraction. If p’ is an isomorphism, X+ = X, x = id. In this case it follows from Lemma 4.1.1, (iii) (v) that on X there exist two t>xtrema.l rays. Then we choose R in that PK # 0. t,lie following commutative diagra,m. Lg

y = TR:x+

g--p-x---*);‘+ ~ 0 -Lx,---“.“/ I I x - - - -c - - * y Definition. center

alon,g

Remark.

C

The diagram

(4.1.1)

(respectkely,

nt x).

Conditions

(*)> (**)

M

will

(4.1.1)

culled un elem,en,twy

be satisfied

ration,ul

if t,he linear

determines a generically finite morphism p: 2 + x a finite nunlher of fibers of positive dimension.

c

ll@““~~~’

map

with

syst,em 1 - K,- 1 having

only

Examples. (i) (M anin (1972)) Let X = X:i, C p4 be a nonsingular cubic and z E X be a point such that there exist only a finite number of lines passing through z (this holds for a general point 5 E X). Then for this pair (X, z) conditions (*) and (**) are satisfied. In this case p: 2 --) X is generically finite of degree two, )7 = z+, and x: X- - - + 2 is the Galois involution of (P (SW 4.1.10). Tl 16, J 1)ase locus of x is t’he union of proper transforms of lines passing through z. The morphism 9 contracts to a point the divisor which is the proper transform of the hyperplane section of X = X:3 c p4, tangent to X at, II: (this is a contraction of type E2). It is easy t,o see that Y z X, i. e. Q: X- ~ +X is a non-trivial bira,tional involution (in (Manin (1972)) this involution was denoted by trr).

(ii) (Kolpakov-P ro kl lorov (1992), X5 c p” be a de1 Przso threefold

Furushima-Nakayama (1989b)) Let X = of degree 5, and let :r E X bc an arbitrary

$4.1. Elrrncntary point. Then a. pl-bundle

Rational

Maps: Preliminary

Results

67

in tht diagra,rn (4.1.1), we have that, Y ‘v l??“: y: J?+ + Y = p2 is (this is a contraction of type C2).

(iii) Let X = X4 C i?’ be a smooth intersection of two yuadrics, and let .z E X be an arbitrary point. Then Y r” p’ and p: %+ i Y = p1 is a quadric bundle (this is a contjraction of t,ype Da). Now, let X be a Fano t,hreefold of index 1 and genus 9. Denote by E c 2 the exceptional divisor of the blow-up 0, by E+ = x(E) its proper t,ransform on x+. Put, H* := o*H, Hf := x(H*), degC := H. C, and let g(C) he the genus of C. Lemma (i)

4.1.2.

Notntion

For un, elementarTy followi’rLg

as ubo~ue.

rutional

map

with

center

along

a CILTIK

C,

one

has

the

relations: -K;+

zzz

-K$

= 29’ - 2,

9’ = g + g(C)

wh,ere

~ degC

- 1,

(-K,-+)’ E+ = (-K,-)” E = degC + 2 - 2g(C), -K,-+ . (E+)” = -K,. E2 = 29(C) ~ 2; E” = -degC+2&2g(C). (ii)

For un elemcn,tar~J th,c following

mtiod

map

with

cen,ter

at

u point

z E X,

ant:

has

relutions:

-K?-+ = -KS = 2.9’ - 2, where (-h’,-+)2. Ef z (-,{--), . E = 4, -“--+ (J3+)2 E -h’,E” = -2,

g’ = g -

4;

E” = 1. Let us check for example

= &gX’

n,L3(-Kx+)” n’(2g

(see 2.2.14).

t,he first, relation. zz ,3(-K,-)”

- 2 - 3 deg C + deg

Tl re remaining

WC have for n >> 0

relat,ions

= ,:$(H*

- E)”

z

C ~ 2 + 2g(C))

can be proved

in a simila,r

way.

Remark 4.1.3. In general it is not, true that (E+)” = E”. We shall call the int,eger e := E” - (E+)” the defect of the elernentary rational map Q. Note that if x = id, then e = 0. In any case e is the “nurnberi’ of point,s in the intersection of E with the excrptjiona,l set for ‘p’, see (Reid (1983a), Tak(~uclri (1989)). Remark

4.1.4. From the exact sequence 0 i

Pit(Y)

L

Pic()T+)

(2’

Z

we get that Pit(Y) y .Z. Therefore Y = p1 in cases Dl, 02, in cases Cl, C2. Moreover: Y is a nonsingular Fano threefold E2, and Y is a Q-Feno threefold in cases E3, E4, E5.

03 and Y = P2 in cases El and

68

Chapter

4.

Fano

Threefolds

with

p = I

Denote hy c?y(l) the ample generator of Pit(Y) in cases El,. . , E5 let D c %+ be the exceptional Lemma (i)

(ii)

(iii) (iv)

4.1.5.

Notation

= Z, L := y*Oy(l), divisor of cp.

and

as above.

If p is of type Cl, C2, Dl, 02 or D3, then L N (1(-K,-+) - pE+, where p = p(R) is the length of th,e extremal ray R (see (1.3.1) and 1.4.4) and 0 is a positive integer. Ifp is of type El, E3 or E4, th,en D N a(-KF,) -rE+, where r = r(Y) is the inden: of the Fano threefold Y (in this case Y is Goresnstein, and 7 is an integer), o E PJ and 0 + 1 E 0 mod r. If cp is of type E2, then D - o(-Kx+ ) - rE+, whesre r = r(Y) is the index of Y, u E W and 2c1+ 1 E Omodr. If y is of type E5, then D - 0(-K,-+) ~ [jE+, where CU,,!~E N> ,O = 2r, r = r(Y) E $Z is the in.dex of Y und Q + 2 E Omodp.

Let, us prove for example the statement (i). It is clear t,hat Z @ E+ . Z. On the other hand, from t,he exact sequence -Kg+ 0 +

Pit(Y)

5

Pic(X+)

(see (1.3.1) and 1.4.4) WP get Pic(x+) This gives us the desired assertion. Lemma Type

4.1.6.

The following

3

Pic(x+)

E

Z ------f 0

= L.ZBD.Z

= f(-Kz++D)[email protected].

hold on x+.

relations

D. L” = L2

(-KY+)

L . (-KF+)~

zz 0,

= (degree

of a general

fiber

of

‘p)

(the last nvumber does not exceed 6 in, case D1, is equal to 8 in case D2 und is equal to 9 in case 03). Type C. L3 = 0 L2. (-Kz+) = 2, I;. (-k,+j2 = 12 - deg A, where A c P2 is the discriml;nant curue P2. In case C2 we put deg A = 0. Type E2. D” = 1: Types

of the conic

bundle

p: x+

D2.

(-Ki-,-+)

= -2,

D . (-K,-+)’

= 4.

D2.

(-K,-+)

= -2,

D

= 2.

E3 and E4. D” = 2,

(-K,-+)2

+ Y =

$4.1. Elementary Type

Maps:

Prelinrinary

Results

E5. D” = 4,

Type

Rational

D”

(-K,-+)

= -2,

D.

(-I(,-+)”

= 1.

El. (-K,-+ + D)2 f (-Kg+) = -K$ = r.degY, (-Kg+ + D) D. (-K,-+) = T. deg r, D” (-Kg+) = ag(r) - 2,

wh,ere the curve r Fano threefold Y. Lemma

4.1.7.

(i) dim ( - rrKx (ii) Furthermore,

:= y(D)

is smooth,

Let n E RI be a positive

and r := r(Y)

integer.

/ = dim 1 - 71h -?+I = A(-Kx)‘n(n if the linear system ( - nKxl

0 --j

O,-+ ( -nKx+

and from the inequality Corollary

+ 1)(2n + 1) + 2n, is base point free, then + 1)(2n

+ l)+

consequence of the R.iemannRoch and the theorems. The inequality in (ii) follows from

+

in Theorem

0%+ (-nKx+) 3.1.3 applied

-

OF;+ (-n.Kx+)

+

0

to (p(E+).

(i) dim 1 - I g’ + 1 + 2g(C) - 4 - deg(C).

g+dC)

(ii)

- E+)

of the

Then

dim 1 - n>Kg+ ~ E+I 2 A(-Kx+)“n(n 271,- (-K,-+)2 . E+ ~ 2. Proof. (i) is an immcdiat,e Kawamata-Viehweg vanishing the exact sequence

is the index

4.1.8.

-dedC)

Corollary 4.1.9. 2 deg(C) + 5 - 3g(C) c c x = x+-2 c center at point x E X

Un,der conditions of Lemma, 4.1.5, assume that g > or an elem,entary ration& map with, center ulong curve f P” +I or g > 10 for an, elementary ra,tionul map with = XzyP2 c IP+l. Then

(i) Q + 1 = r i7~ case (ii) of Lemma 4.1.5 pi. e. El, E3, (ii) Q = 1, r = 3 in case (iii) of Lemma 4.1.5 (i. e. E2), (iii) cv = 1, p = 3, r’ = 3/2 or (Y = 3, ,!? = 5, r’ = Lemrnu 4.1.5 (i. e. E5). Proof. From 4.1.7 and 4.1.8 we have dim / - Kx+ D is fixed.

- E+l

E4), 5/2

in case (iv)

2 1, but the divisor

of

70

Chapter

Remark

4. Fano

4.1. IO. If the linear

Threefolds

system

with

( ~ nK,-

p = 1

(, n E N; determines

a gener-

ically finite morphism of degree two @j:2 -+ X c IP”, then )7+ 2 2, x is a birational Galois involution for (p, hence Y N X. However. in general it, is not true that x is an isomorphism. Now, we recall some general facts about Fano t,hrecfolds. Let X be a Fano threefold with Gorenstein canonical singularities. Assume that the anticanonical linear system I- Kx j determines a morphism (PI-K~ 1: X --f lP+‘, where .y is the genus of X (note that by 2.4.6 this assumption holds if X is nonsingular is and p(X) = 1). Then by Th eorem 3.1.12 there are t,wo cascls: either wlPKxi an embedding or ~1 P&-X I is a finite morphism of degree 2 onto its image which is a. va,rict,y of minimal degree (see Theorem 2.2.11) in lP+’ In t,he latter ca.sc X is said to be a hyperelliptk Fano threefold. Hyperelliptic Fano threefolds are easier to classify (see 2.1.16). In particular t,he following result holds. Proposition 4.1.11 (Iskovskikh (1979a), Iskovskikh (1988)). Let X be a nons~ngulur Funo threefold of genus g with p(X) = 1. Th,en the linear system, 1 - KxI determines an em,bedding X = Xzy-z c-i IP+’ of X a,.~ a subaariety of degree 2g - 2 except for the foltowin,g ho cases: (i) ~~~~~~~ 1: X + of degree

(ii)

yj-Kx

P3 is a ,firvite

morph,ism

of

degree 2 ram$ed

ulong

1: X

quad&

4

Q c IP”

Q mm$ed

Now, we consider

is a finite

morph,l:sm

u surface

along

the case when

2 onto

of degree

of dqree

u nonsin~gular

8, g = 3.

I - Kx 1 d&ermines

an embedding.

Proposition 4.1.12 (Iskovskikh (1979a), Iskovskikh (1988)). Funo th,reefold of genus g with canonical Gorenstein singularities. I ~ Kxl determines un em,bedding X = XzcI/-2 -+ IP+l. Then (ij (ii)

th,e kuge X2yP2 is a projectively if g = 3, th, en X = X4 C P4 iS un inter-se&on of a quad+ and a complete

(iii)

if

X

in,tersection

g 4

embedding norrrLa,l

4.1.13. 5 with

p(X)

X = Xz1--2 variety

a

p(X)

=

cub&

variety, if g =

Let X be u Assume that

4, then X = X(j C P then X = X8 c P”

if g = 5,

iS

is

quadrks, 1

and 9 > 5, then

X

%~ 2 c

iPI+’

is

an

quudrics.

of

Corollary

normal U CplUTtiC,

of three

*is nonsingular,

intersectaon

genus

a surjace

6, g = 2,

wh,ich

Let X be a nonsingular = 1. T&n the linear ~-i P ‘Jfl, und the image is an intersection

system

Fano

threefold of index 1 and I - Kx 1 determines an

Xzrl-2

c

Pfl

is a projectisuely

of quudrics.

Remark 4,1.14. Fano threefolds X such that the linear system I - Kx I determines an embedding whose image is not an intjersection of yuadrics are called trigonal (cf. Theorem 2.2.9). They can be characterized by t,he property that a general curve-section XzyP2 n P”-’ is a canonical trigonal curve. A complete classification of nonsingular t,rigonal Fano threefolds was given in (Iskovskikh (1979a)j, see also (Iskovskikh (1988)).

54.2. Farnilirs

54.2. Families

of Lines and Conies on Fano Threefolds

of Lines

and Conies

71

on Fano Threefolds

Later throughout this chapter, if we do not specify the opposite, we shall assurlre that X = Xc J!l -I 2 C IW+’ is a nonsingu1a.r anticanonically embedded Fano threefold of index 1 and genus g > 3 with p(X) = 1. Recall that there exist only t)wo types of Farm thrcefolds X of index 1 with p(X) = 1 such t,hat the anticanonical linear system j ~ h’x / does not determine an embedding (see 4.1 .ll). W7e shall also assume that on X there exist, at least one line and at least one conic. Proofs of existence of lines and tonics will be done in Sect. 4.4 and Sect. 4.5, respectively. Lemma Xzgp2

4.2.1 c IP+’

(see for example Iskovskikh br: an arbitrary line. Then Nclx

(0, -1) (1. -2) (ii)

NC/X

(1989h)). (i) Let C c X onw of th,c following h,olds

= Up1 (I> O,I (-1); = C’p (1) CE0,1(-2),

Let C C X = X2yp2 C Pq+l be a non-de,yen,emte g > 5. Then, one of th,e ,following holds NC/X

(O,O)

Nclx

(2, -2)

=

and assume

conic

that

= UPI c~i C’pi 3

= &l(2)

63 Qml(-2),

1~ a hyperplane section of X. We prove for example (ii). Let H N -Kx Let c: X + X be the blow-up of C. Let, H* := c*H, and let E := a-‘(C) be the exceptional divisor. Then E is a Hirzebruch surface F,, for some e > 0. Consider t,he linear system A c 1H 1 of hyperplane sect,ions passing through C. Since X = X2,q-2 C pD”+’ .1s an intersection of quadrics (see 4.1.13) and Pit(X) = Z H, we have C = X f’ (C), where (C) is the linear span of C. Therefore the linear system A cuts out on X the curve C as a scheme and lH* - E( = a* A has no base points. Now: we claim that A contains a nonsingular surface. Indeed, in the opposit,e case a general surface F E A is singular, but by Bertini’s t,heorem its proper transform F’ E a*A is nonsingular, i. c. each curve from the restrict,ion fl*Alfs contains a fiber of E = IF, as a component,. Since (drills is not, composed of fibers, this contradicts Bertini’s theorem. Therefore there is a smooth hyperplane section passing through C, a K3 surface H E A. Then t,he normal bundle of C is an extension 0 +

NC/H

+

NC/X

d

where NC/H = 0,1(-2) and NH,xlc; 0,1(-d), where (I = 0; 1 or 2.

=

NH/XI 3 with p(X) = 1. Assume that on X there exists ut least one line. Denote b;y r = r(X) the Hilbert scheme that parametrizes lines on X. Then (i) r is of pure (ii)

dimerkon

if t E r is an arbitrary

1,

point

and C = Ct is the corresponding

line

on X,

then

(iii)

(iv)

u) t is nonsingular Zf and only ,if Nc:/x is of type (0, -1) (see 4,2.1), b) t is singular if and only Zf NC/X is of type (1, -2), an irreducible component r’ c r is non-reduced at a generul point if and only if NC/x is of type (I, -2) f or all lines which correspond to points from P; this 1;spossible only in the following two cases: a) g = 3 (i. e. X = X4 c I?4 is a smooth quark:), and the ruled surfuce R’> swept out by lines from S, is a cone over a smooth plane cu.rve of de,yree 4, b) R’ is swept out by projective tangent lines to some cuwe B’ c X, for g > 4 there are at most 6 lines passing through any point x E X, for g 2 5 there are at most 4 lin,es passing through uny point T E X.

Remark 4.2.3 (Zskovskikh (1977)). Note that in the case 9 = 3 there are Fano threefolds X = X4 c p4 that contain cones, for example, the Ferrnat quartic C:=,, 2,4 = 0 In this case all the lines lying on cones have normal bundles of type (1, -a), see (Tennison (1974)). However, a general quartic X = X4 c p” does not contain cones. Some assertions in Proposition 4.2.2 are easy consequences of deformation theory. Indeed, if C is of type (0, -l), then h”(Ncix) = 1, hl(Nclx) = 0. Therefore at the corresponding point t E r, we have dim Tt,r = dimt r = 1: i. e. in this case Proposition 4.2.2 is proved. Now, let C is of type (1, -2). Then ~,“(Nc,~) = 2 and hl(iVc,x) = 1. Whence dirnTt,r = 2 and diml r > 1. to show that the sit,uation when To prove (i) and (ii), it 1s sufficient dimt r = 2 and r is nonsingular (and reduced) at t is impossible. Ohviously, this is an open condition, and since dim Tt,r < 2, t,his condition also is closed. Therefore if this condition holds at some point t E r, then it holds at, all points of some irreducible component of r. Let P = P(X) be the (rernoved) family of lines on X. Let P-X

r red

I’

s4.2.

Families

of Lines

and

Conies

on Fano

Threefolds

73

be the diagram of natural morphisms, where n-: P --i rsed is the corresponding @-bundle, and pl,-~(~):~-‘(t) + Ct c X is an isomorphism for any point t E rvxi of dirnension 2, let P’ := T-‘(T,‘~,~), Now, let r,!ccl c r&l be a component mce dim r’ = 2, we have dim P’ = 3 and dim R’ = 2 and let R’ := p(P’). S’ or 3. If dim R’ = 3, then R’ = X and the morphism p: P’ + X is generically finite. We derive a contradiction in this case. Indeed, for any point, t E r, we have a natural morphism of normal bundles dp:N,-l(,)/p

If t E r’,

----f

Nzr/x.

then NT-y,)lp

= Op

~3 &I

and Nzt/x

= 0,1(l)

@ 0p1(-2).

Since Hom(&l, c?pl(-2)) = 0, th e morphism dp has at each point t E P’ a kernel of dimension at least 1, and by Sard’s lemma p: P’ ---) X canriot be surjective (here we use our assurnption about the charact,eristic of t,he ground field). Therefore dim R’ = 2. By our construction the surface R’ contains a two-dimensional fa,mily of lines. But there exist,s only one projective surface with this property, narnely p2. Since Pit(X) = ;Z. H; this case is also impossible. Thus we can have only possibilities (ii), (a) and (b) In the situation of (iii), let; as above, r,& and P’ be the corresponding closed components with the reduced structure. Then r:G,cl is an irreducible curve with at most a finite number of singular points. For a general point t E r:ed \ Sing(ric,l), t,he cornposition of the differential dp:

N+(t)p

+ II

61

+

Nzt/x

II OP’(1)8 c&1(-2)

with the projection on the second surnrnand is the zero-morphism. Therefore on a fiber Y* (t) there is a urlique point st at which the morphism 0~1 + c?pl (1) has a non-zero cokernel. When t va.ries in $I,,, the point st varies in a curve A’ c P’, a section of 7r: P’ + r,!ed. At each point st E A’, t,he differential of p: P’ + R’ c X has a two-dimensional cokernel. We have only the following two possibilities: (a) For almost all st E A’ c P’, a tangent vector to A’ at st is contained in the kernel of the differential dp. In this case the morphism p: P’ + R’ c X contracts A’. Hence R’ is a cone contained in X c pgfl. Then R’ is contained in every tangent hyperplane to X at 20 E X, the vertex of R’. Such hyperplanes cut out on X a linear system of dimension g - 3. If g 2 4, then R’ is a fixed component of this linear system, which contradicts our assumption that Pit(X) = Z. H. We obt,ain case (iii), (a,). (b) For almost all points st E A c P’, the irnages of tangent vectors t,o a fiber of P’ and t,o the section A’ c P’ are proportional in T,(,s,),~.

74

Chapter

4.

Fano

Threefolds

with

p = 1

In this case the corresponding lines Ct; t E $I,,, are tangent, to the curve B’ := &A’) at p(st) E X, i. e. we have case (iii), (1,) (this case was missed in (Iskovskikh (1978)) and also in t,lie proof of Proposition 2.1. Chap. 3 in (Iskovskikh (1979a)). Remark 4.2.4. (i) It’ follows from the proof of Proposition 4.2.2 t,hat, all the singular points of the curve r are of embedded dimension 2, i. e. they have two-dimensional Zariski tangent, space:. 6i) Assertions (i)> (“)n and also (iii) (except for the condition y = 3) of Proposition 4.2.2 rcrnain true if in&cad of Pit(X) = z H, WC assume only the absence of projective planes p2 on X = X2:,-2 c p”+’ (Iskovskikh (1979a). Iskovskikh (1988)).

ded on

Proposition 4.2.5. Let X = Xzyp2 Funo threefold of index 1 and g&s X

thee

exists

a no,wdegenerate

cor~ic.

c pgfl g >

he un anticnnonicall~y 5 with

p(X)

=

1. As.ss~rnr

ernbedthat

Then

(i) X contains a two-dimensional family of conks. (ii) Through alm,o.st every poind n: E X there passes a ,finite (iii) A general conic C on X is of type (0,O) (see 4.21).

n~umber o,f tonics.

Proof. Let r = r(X) bc the Hilbert scheme parametrizing tonics on X (including reducible and non-reduced), let P = P(X) be the corresponding removed family of lines with base r, and let 2) P-X

be the diagram of natural maps. Let r’ c r be an irreducible components and let P’ c V’(F) be the corresponding family over I”. By deformation theory and Lemma 2.2.1, we have dirnr’ > h”(NCY,x) - !L~(N,,,,) = 2, where C is a general non-degenerate conic frorn P’. We have to show t,hat a general point from the family P’ llas type (0,O). A ssume the opposite. Then t,he differential

has at least, a one-dimensional cokernel Opl (-[I); (-1 = 1,2. In particular, the map p: P:c,,l + X is not, surject,ive. Therefore R := p(P:c,,i) is a surfa.ce in X. The surface R conta.ins at least a two-dimensional family of tonics and, according to the classical result,, see (Iskovskikh (1979a), Iskovskikh (1988)), it is either the Veronesc surface J’d C lF’ or it,s projection to a lower-dimensional subspace. Whence dcg R 5 4. But by our assumption Pit(X) = Z H and g > 5: so X tlo~ not, cont,a.in a surface of degree < 8.

Ij4.2. Families

of Lines and Conks on Fano Threefolds

75

Thus we have shown that for a,n irreducible component r’ c r; a general conic C from the corresponding family P’ has normal bundle Nc:lx = 0~1 @ Opl, and t,he rnorphisrn p: P’ 4 X is surjective. We havcx /L”(N~;,~) = 2 and h’ (N& = 0: hence, by deformation theory, dim r’ = 2, and the scheme r’ has no singularities (in particular is reduced) at a gcncral point. Therefore dim P = 3, and P also has no singularit,ies at a genera.1 point,. Sinc~c the map p: P’ --f X is surjective, it is generically finite, i. e. there are a finit,e number of tonics (from t,he family 1”) passing through alrnost every point 2 E X. Since the schemes r and P contain only a finite number of irrcduciblc components, the assertion above also holds for all tonics on X. This proves Proposition 4.2.5. Lemma

4.2.6

(see Takeuchi

(i) For g > 10 t h ere are arbitrary

(ii)

point

I(: E X

for g ) 9 every n,umber

(iii)

of

conk

(1989)).

at most

a ,finite

= X29-z

c iFg+‘,

C X

= X2,q-2

C

Notatio7t,s

num,her

as

aboue.

of tonics

passi71g

intersects

ut

C iF+’

through most

an,

a jiflvite

lin,es,

fo7. g > 5 0, ,fi7vite rlumber

generul

conic

C C X = XZ!~-~

C

p, li+’

intersects

at

m,ost

II,

of lines.

Proof. (i) Assume that there is a one-dirncnsional family of tonics passing through c t X. Let F be a surface swept out, by these tonics: and let A = IH3rl C /HI be the linear system of hyperplanc sections of X having a. singularity of multiplicity 2 3 at point 2. Thc>n dim fl > dirn lHl - 10 = g ~ 8 > 0. Each conic passing through X is contained in every divisor frorn fl. Therefore F is a fixed component of A, a contradiction to Pit(X) = Z . H. (ii) Simi1a.r t,o (i). (iii) According to Proposition 4.2.2, all lines on X sweep out a divisor R on X. It follows frorn Proposition 4.2.5 that a general conic on X is not contained in R. Then for every point frorn the intersection C n R = (21. : z,}; there are only a finit,e number of lines passing through it. Families of lines and tonics are importa.nt, objects for investigat,ion beca.use properties of such farnilies reflcrt geometrical propert,ies of Fano varieties themselves and can be used for computing intermediate .Jacobians of Fano thrccfolds and also for st,udying the groups of birational aut)ornorphisms (see Chap. 8 and 9). Below we collect the simplest properties of farnilies of lines on nonsingular Fano t,hreefolds, see (Tennison (1974). Markushevich (1981)) Puts (1985): Bloch-nlurre (1979): Ba.rth-Van de Ven (1978). Collino (1979a)). Theorem

4.2.7. Let X be n Fano thrrefold of i7dx 1 and ,genus g mith Let r = r(X) he th, P curve th,at pnmmetrizes lin,es on X, and let R = R(X) be the corresponding swfcm: swept out by lirres on X. Th,en for u gen,erd (i7~ thr> sen,se of the m,oduli space) variety X the curue r(x) is irreduciblr, reduced ard r~,on,singular. fin, this situation the curve T(X) nn,d the Pit(X)

=

Z.

76

Chapter

surface R(X) the genus g):

4. Fano

have the following

9 g 9 Y Y 9

= = = = = =

g =

discrete

g(r) g(T) g(T) g(T) s(r) g(r) g(r) g(T) g(T)

g = 2,

3, 4. 5, 6, 8, 9, 10,

g = 12:

Threefolds

= = = = = = = = =

6865, 801, 180, 129, 71, 26, 17, 10, 3:

with

numerical

p = 1

invariants

R R R R R R R R R

(depending

on

- -624Kx, ,-a -8OKx, N -30Kx, - -lBKx, - -lOKx, N -5Kx, - -4Kx, N -3h;i, N -2Kx,

In the cases g < 6 and g = 8, the variety X is isomorphic t,o a complete intersection in a weighted or usual projective space or into a Grassmannian. Then the scheme r(X) can be given as the zero locus of a section of some vector bundle. For g > 9 it is possible to use the method of double projection from a line 4.34, see (Prokhorov (199Oc)), 4.3.9 (iii) and also 5.2.14. Families of conies on Fano threefolds were investigated by Tennison (1974), CollinoMurre-Welters (1980), Letizia (1984) for g = 3 and Logachev (1983), Iliev (1994b) for g = 6 ( see also Remark 5.2.15). The case g = 2 was considered by Ceresa-Verra (1986). Remark 4.2.8. Since Fano threefolds X of index 1 and genus g 5 5 with p = 1 are complete intersections in (weighted) projective spaces, the paramet,er count gives us that there exists a line and a conic on X; see for exarnple (Tennison (1974), Collino (1979a), Collino-Murre-Welters (1980)) for the ca,se g = 3.

$4.3. Elementary

Rational

Maps with Center along a Line

In this section we shall follow t,he works of Iskovskikh (1979a), Iskovskikh (1988)? Iskovskikh (198910) and Cutkosky (1989). Let, as above, X = X2$,-Z c p”+i be a nonsingular anticanonically embedded Fano threefold of index 1 and genus g > 4 with Pit(X) = Z H. We assume that, on X there exists a line. The existence of a line will be proved in Theorern 4.4.13. Proposition 4.3.1 (Iskovskikh (198913)). Let X = X2cI--2 c P+‘, C c X be a line. Let o: X + X be the blow-up of C, H* = a*H and let E = o-l(C) be the exceptional divisor. Then the linear system / - K,-1 = IH* - El is base point free and determines a morph,ism p: j;; i

x c py’+l,

g’ := g ~ 2,

5 (the case g = 4 can be treated in a similar way). First, we look at fibers of (p not contained in E. The preimage of any point, y E x under the projection 4,: );; - - +X C IIDg+l is t,he intersection Xnp” Y' where p” Y is the plane in IVJ+’ passing through the line C and containing the point y E IV-‘. Since g > 5 and Pit(X) = Z. H, then X is an intersection of quadrics (see 4.1.12). Therefore X n Yi is an intersection of tonics in I$ because 8’)’ $ X. That is why only the following cases are possible: a) b) c) d)

X fI p2 = C + {point of multiplicity l}, X f’ p,y = C + Cl, where Cl c X is a line intSersect,ing C, X n p, 9- - 2C is a, double line, i.e. pi tIouches X along C, X”iD$=C+?,,: where 11~ is a tangent vector at some point

x E C c X.

Since the family of lines on X is one-dimensional, then the lines intersecting C cannot cover the whole of X (Proposition 4.2.2). Therefore we have tha.t case a) occurs for ahnost all points y E X, the map 4: X - - + 2 is birational, and non-trivial fibers of ,(c,are lines intersecting C. Further, the ruled surface E = p(N c/x) is isomorphic either to F1 if NC/X is of type (0, -1) or to IF3 if NC/X is of type (1, -2). The restriction I- KFIE has the form s + af, where s is the class of the exceptional section, f is the cla,ss of a fiber, and LY E Z. The coefficient Q can be found from the equality

78

Chapter

4. Far10

Threefolds

2 + 2cr = (s + afy

wit,h

= (-x,-y

p = 1

E = 3

(see 4.1.2). Hence n = 2 in case (0, -1) and (1 = 3 in cast (1, -2). We gclt that the linear systcrn j - Kxl~ /s + (jfl d ocs not, contract curves in case (0: -1) and contracts only the except,ional section s in case (1, -2). We investigate now t,he variety x PO+’ and choose a nonsingular h’3 surface S E / - K,-1. From the exact sequence c

c

0 -

u,-

-

“,+Kg)

-

c&y-K,-)

+

0

we get t,hat the restriction 1~ “2 1,y is a complete and base point free linear system. According to the classical result, about h’3 surfaces (Saint-Donat (1974)), t,he image p(S) lPq is a projective normal K3 surface with at worst Du Val singular points (we arc considering t,he case g > 5, i. e. @ is hirational). On the other hand, p(S) is a hyperplane section of the threefold x = q(X) c IPtl. From this we can get that x IP+’ is projectively normal (see 2.2.4). Further, by the construction, -2T_;r = $5*0,(l), and t,he restriction @: X0 + X0 c

c

to some open subsets z-,, X and X0 2 with codim(X - X0) > 2 is an isomorphism. Then from the inequality -h’,,) = U,(l)lx,,, it follows that -K, = U,(l), i. e. -h’, IS a Cartier divisor, and K,- = $*K,, i. e. X has only canonical singularities. But the canonical singularities of index 1 arc Gorenstein. see (Clemens-Kolllir-Mori (1988)). If p c1oes not contract divisors, then x has only terminal singularit,ies. It remains to prove the finit,eness of the number of fibers of type (i). Suppose that t,he rnorphism $? contracts some surfa.ce B onto a curve. Obviously, B N aH* ~ [3E for some 0, p E W. Since the surface n is contracttd by the morphism p, then c

0 = (-K,_)’

c

B = (H* - IS)’

(aH* ~ /jE) = 42y - 2) - (N + 2/Y) ~ @,

whence

we have /I = a( ig ~ 1). C onsidcr

surface

E

E I - K,-1. Then

B);7

= Ce,,

the restriction where

curves, the fibers of (p (a.ccording: to the facts proved transforms of lines intersecting C). Wc have CT!% t B = (CL ’ (C

i”,))i,

= (iT~)lk

+ C(E?, .I#,

of L? to a general

C, 7, are smooth above,

Ej)lg

rational

e, are proper

= -2,

where in the last sum (Et eJ) /g = 0 1)ec:ause line CII intersecting C;: and C must lie with them in the same plane ‘P’. This contradict,s t,he fact t,ha,t X is a,n intersection of quadrics. On the ot,her hand,

Whence we get p = N + 2 = a($ ~ l), i. e. (29 - 6)” = 6. If g > 5, then the last equality is possible only in the case when g = 6. CY= 1, and lj = 3. We shall calculate in t,his case the degree of the curve 2 := $5(U):

$4.3. Elementary

Rational

Maps with

Ccntcr

along a Line

79

degZ=B+K$H*=(H*-3E).(H*-E).H*=7. From Proposition 4.1.12 we get that X = X~j c p” is t,he complete intersection of a quadric and a cubic. The surface E = q(E) is of degree 3 and its linear span is a hyperplane p” c P”. The curve Z: evidently, is contained in E, and t,he variety x is singular along Z. Let x fl (E) = E + El, where (J!?) is the linear span of E. Then El is a (possibly reducible) surface of degree 3. The divisor X n (E) = J!?+ El is singular along Z, so Z c En El. If E # El, then there exists a quadric Q c p4 that contains E and does not, contain El, i. e. Z c Qn&. a contradiction with deg Z = 7. Therefore E = El. and (E) = l?” cut out on X = X6 c p” an irreducible divisor 21!?. But then writing down equat,ions of a quadric and a cubic defining X, it is possible to prove that dimSing > 2. The last contra.diction proves Proposition 4.3.1. Thus, we proved t,hat any line on the Fano threefold X = Xzgp2 C pY+’ satisfies conditions (*), (**) of Sect. 4.1. Now let us consider an elemcntary rat,ional map with cent,er along a line C. The following theorem was proved in (Iskovskikh (1979a), Iskovskikh (198X), Iskovskikh (1989b)), see also (Cutkosky (1989)). Theorem 4.3.3. Let X = XZ,~-~ embedded Fano threefold of index 1 H := -Kx, and let C c X be any contraction of the extremd ray cp = following possibilities: (i)

(Iskovskikh

c p- q+’ be a nonsingular anticar~onically und genus g > 4 with Pi X16,

Proof. lows that ities:

of X are contain,ed x14

14

in the following

Xl6

5

c F+‘,

X18

3

then: table

X22

2

0

l.!Z.s), and X18 and X2;? are rational.

From the classification of extrernal rays (see Theorem 1.4.3), it folfor the contraction y: X+ + Y, one has only the following possibil-

Type D. Y = PI, and L - a(-Kg+) - pE+, where p = p(R) = 1,2 01 3 in cases Dl, 02 or 03 respect,ively (see 4.1.4 and 4.1.5). Using 4.1.6 and 4.1.2, we get 0 = L’. (-K,-+) = (2g - 6)a2 - 6~1-1 - 2p2, (the degree of fiber) = L. (-K,-+)2 = (2g - 6)~ - 31~. Solutions

of these Diophantine

Type C. Similarly where p = p(R) PE+ >

equations

gives us case (iv).

to t,he previous case, Y = IP’, and L N a(-K,-+) = 1 or 2. F’rorn the Diophantine equations

2 = L” . (-IT,-+) 12 ~ degA = L

= (29 - 6)~” - 6cup - 2p2, (-A-,-+)” = (29 ~ 6)0 - 3p ,

we get cases (ii) and (v).

Types E-ES. 4.1.2, we have

D N a(-K,-+) -2 = 0’. (-I$+) y = D (-K,-+)’

where

- /?E+,

a,P

= (2g ~ 6)0’ - 6n,8 - 29. = (2g ~ 6)~ - 30,

y = 4,2 or 1. We get the Diophantine

equation

a/32 ~ 2 = a(y - a/?) without

solutions.

> 1. By Lemmas

So these cases are impossible.

4.1.6 and

34.3. Elerncnt,ary

Rational

Maps with

Center along a Line

Type El. By Lemma 4.1.5, D N a(-K,-+) -rE+ where a > 1 and T 5 4. Using Lemmas 4.1.6 and 4.1.2, we obtain r3 degY = (-KY+ + D)2 -6r((u + 1) - 2r2,

rdegr

= (-K,-+ + D) -3r(2a + I) ~ 2r2> 29(r)

D.

a+1

(-Kg+)

= (2g - 6)(a

+ 1)2

(-K,-+)

= (2g - 6)(0

+ 1)~

- 2 = (2g - 6)a2 - 6r(u - 2r2.

81 = Omodr,

(4.3.1)

(4.3.2)

(4.3.3)

First, let g > 7. Then from Corollary 4.1.8 we get a = r’ - 1 and T 2 2. Equation (4.3.1) can be written rdegY = 2g- 14. For T = 4 and r’ = 3 we get cases (vi) and (vii), respectively. The degree and the genus of r are computed from (4.3.2) and (4.3.3). The fact that the curve r is not hyperelliptic in case (vi) will be proved in 4.3.9. In the case T = 2 we have 1 5 degY < 5 (see 3.3.1). Equation (4.3.1) can be written 9 = degY + 7. From (4.3.3) we get g(T) = 9 - 12 = degY - 5. That is why the only possible case is when degY = 5, g = 12, g(r) = 0 (i. e. (viii)). Let now g = 6; then by Corollary 4.1.8 we have / - K,-+ - E+ 1 # fl and t,his is why D N -E’, T = 1 or Q < T, a+ 1 = Omodr, 7’ > 2. From equations (4.3.1)-(4.3.3) we get t,hat only the first case is possible, and Y = Y~o c p7, i.e. we have (iii). Case g = 5 can be treated in a similar way. The difference is that from 4.1.8 it follows that (2 < 2r. In case g = 4 we have x = @’ and therefore we can use Remark 4.1.9. The theorem is proved. Remark 4.9,s. One can compute that in each of the cases (i)--(viii) the defect e = E” - (E+)” rational map $: X - - + Y ( see 4.1.3) of the elementary is not zero. Therefore t,hr exceptional set of the morphism y’ is non-ernpt)y. Therefore if C is of type (0, -l), then there exists a line on X intersecting C. This is not true in the case (1, -2) (see 4.3.8). It follows from Theorem 4.3.3 that the genus g of a nonsingular Fano threefold with Pit(X) = Z. (-K x ) can only be 2 < g < 12, and g # 11. Later we investigate the question of the existence of such threefolds of genus g = 9,10 and 12. It turns out that all these threefolds can be constructed by using the construct,ion inverse to the one applied in Theorern 4.3.3 (the “double projection” from a line). The following theorem was proved in (Iskovskikh (1979a), Iskovskikh (1988), Iskovskikh (1989b)) for a general curve r and in (Prokhorov (199Oc)) for an arbitrary curve. Theorem 4.3.7. (i) Let r c Y = Yc, c p” h,e u smooth, irreducible CILTIJ~ of degree 5 and genus 0 lying on a nonsingulnr de1 Pezzo variety of degree 5 (see 3.3.1). Th,en l- satisfies Ihe wnditions (*), (**) fro,m Sect. 4.1,

82

Chapter

r

through m,entary

passes ra,tional

nonsingular

with C

Pit(X)

C

X

l(13YC3)

(ii)

(iv)

a unique mo,p

Threefolds

hyperplane

$1: Y - - +X

with

section ,with

center

p = 1

F = F(I’), along

r

and the elemaps

Y

onto

a

Fun,0 threefold X = X22 c P1” of index 1 and genus 12 = -K x I Z. The map 1// contracts the surface F onto a line and %s determined by the linear system e. it is of type El) W.

Let r c & c P4 be a, smooth irreducible curue of degree 7 and gen’us 2 lyin,g on a smooth quadric. Then r satisfies the conditions (*), (**) from Sect. 4.1, th,rough r passes a unique krreducible surfuce F = F(r) c IO,(a) 1 und the elemen,tary rationa, map $,: Q - - +X with center alony r maps the quadric onto a nonsingular Fan,0 th,reefold X = Xl8 c P’l of index 1 and genus 10 with Pit(X) = -Kx . Z. The map $ contracts the surface F onto a line C c X (i. e. it is of type El) and is determined by the

(iii)

(i. -

4. Fano

linear

system,

I(Oy(5)

- 2rl.

Let r c P” be a smooth irreducible non-hyperelliptic curve of degree 7 aand genus 3. Then r satisfies th,e conditions (*), (**) from Sect. 4.1, through r passes a unique irreducible cubic surface F = F(r), and th,e elementary map $1 p3 - - +X with center along r *maps p:’ onto a nonskyular Fano th,reefold X = X16 c P1’ of in,delc 1 and genus 9 *with Pit(X) = -Kx . Z. The map contracts the surface F onto a line C c X ci. e. it is of type El) and is determined by the linear system ~(O,:S (7) - 2r/. Constructions described in (i) ~(“‘) m are inverse to the corresponding constructions of Theorem, 4.3.4. Every nonsingular Fano threefold X = XQ--2 c iFJ‘+I of index 1 and genus g 2 9 with Pit(X) = -Kx . Z can be obtained by one o,f the constructions described in (i) -(iii).

The roof of Theorem the group Pic(L+).

4.3.7 is similar

to the proof

of 4.3.4: computations

in

Example 4.3.8. (Iskovskikh (1988)) ( see also (Furushima (1992)) Let Y = Yc, c p” be a de1 Pezzo threefold of degree 5, and let S c Y = Y? c IID” be the SLz(C)-’ mvariant surface which is cut out on Y by a quadric, (see 3.4.9). Then t,he normalization of 5’ is isomorphic to p1 x P’, and the map V: P1 xi?’ + S c p” is determined by an (incomplete) linear system of bidegree normal curve (1,5) (see 3.4.9). On th e surface S c p” there cxist,s a rationa. r of degree 5, the image of a ruling of p1 x P1 of bidegrre (I, 0) under the morphism V. Such a curve is unique up to the group action of SLz(@I). In this case diagram (4.1.1) for t,he elementary rational map with center along r can be completed as follows:

$4.:3. Elementary

Rational

Maps with

Center along a Line

83

(4.3.4)

x=

X‘J‘J c IP13 --?il-

The exceptional locus of the morphism rational curve Z+, the proper transform tangent to F. Such a, curve has normal

+

Y=

Yr, c P”

cp+ here consists only of the smooth of a line lying on S and which is bundle NZ+,v = C$ @ C$l(-2).

That is why the flop X:X + ~~ + Y with center along Z+ is the “pagoda of M. Reid”: see (Reid (1983a)). S’lm pl e computations show that “pagoda” (4.3.4) has only three levels. In diagram (4.3.4) the morphisms ~1, ~2, (pi, (~2 are blow-ups of smooth rational curves with normal bundles of type C&M $ OIFl (-a), and the morphisms ~3, (~3 are blow-ups of smooth rational curves with normal bundles of t,ype C&(-l) :+j 0~1 (-1). The morphisms CJ~ and (~3 have the same exceptional divisor which is isomorphic t,o l?l x l?l and can be contra,cted in two different directions. The variety X = X22 c pi” obtained in this way is unique up to isomorphism. It was constructed in (nIukai-Urncrllur~~ (1983)) using different arguments (see 5.2.13). Remarks 4. t8.9. (i) One can derive the existence of curves from Theorem 4.3.7 by choosing them on a nonsingular surface F (in this case F is a de1 Pezzo surface of degree 5. 4 or 3 if g = 12, 10 or 9, respectively). Then the curve F has exactly five 4-secant, lines if g = 9, four S-secant lines if g = 10 and three 2-secant, lines if g = 12. Those lines lie on F, their proper t,ransforms on y have normal bundles of type Opl(-l) @ (3~1(-1) and are centers of the flop X, see (Iskovskikh (1979a)). The Weil group of de1 Pezzo surface F, see (Manin (1972)), acts on the set of representatives of curves F in Pit(F) transit,ively (Prokhorov (199Oc)). (ii) Let F C p” 1)e a smooth cp: $

+ p’ be the blow-up

(a) F is hyperelliptic, (b) F has a 5-secant line; is not numerically (cl -qz Therefore condition

irreducible of F. Then

curve

of degree

the following

7 and genus 3, and let,

statements

are equivalent,

effective.

a hyperelliptic curve F c p’ of degree (*) of Sect. 4.1.

7 and genus 3 does not satisfy

84

Chapter

4.

Fano

(iii) For t,he surface (Nagata (1960)).

F from

Theorem

Threefolds

with

p = 1

4.3.7, one has the following

possibilities

(a) F is a de1 Pezzo surface with at worst Du Val singular points (see example 2.1.5 (ii)), (1,) F has a line of singular points, and it is the project,ion of a nonsingular surface F of minimal degree (see 2.2.11). There is a good crit,erion relating and the type of the surface F: t,he line C is of type the line C is of type Based on this criterion, following staternent:

the type of the normal

(0, -1) (1, -2)

+=+ ++

the paramet,er

the surface the surface count

(Prokhorov

bundle

is of type is of type (199Oc))

of the line C

(a), (1~). proves

the

Fano threefolds X = X 29- 2 c l?“+’ of genus 9 > 9 containing a line of type (1, -2) f arm a subset, of codimension 1 in the moduli space of Fano t hrecfolds. As a consequence we get that a gmeral (in the sense of the moduli space) Fano threefold X of genus g > 9 does not, cont,ain (1, -2)-lines. So the curve parametrizing lines on X is reduced, irreducible and nonsingular. -. 2 C p”+l (iv) (Mukai (1992a), Prokhorov (199Oc)) S’mce lines on X = X ‘31 form a one-dimensional family, the embedded curve r c Y from Theorem 4.3.7, roughly speaking, is not uniquely determined by the variety X. Ncvcrtheless, the curve r is uniquely defined by the threefold X up to isomorphism. This follows from the Torelli theorem for curves since the Jacobian of r is isomorphic to the intermediate Jacobian of X. Fano threefolds X = X2y-2 c IW+’ lines is not reduced at a general point (Iskovskikh (1978), I 5,k ovskikh (1979a)), that for g 2 4 such threefolds do not structed an example of a Fano threefold lines is not reduced at every point (see

with p = 1 on which the family of are of special interest. In the works the first author erroneously assumed exist. Mukai-Umemura (1983) conXtirr C PI” ] on which the family of 4.3.8, 5.2.12).

Proposition

4.3.10 (Prokhorov (1990b)). Let X = X21-2 c IF”+’ be a Fmo threefold of genus g = 10 or 12 with p(X) = 1 such that the family lines on X has a component ‘not reduced at the general point. Then g = 12, and X = X&lJ zs the Mukai- Urnernura threefold. nonsingular

The variety X2,‘l” also has another interesting properties, group of biregular automorphisms Aut ( Xkin21u ) is isomorphic 5.2.13).

for example its to PGL2(@1) (see

34.3.

Elementary

Rational

Maps

with

Center

along

a Line

85

Proposition 4.3.11 (Prokhorov (199Oa)). Let X = Xzrl--2 c p”+’ be a norxingular Fano threefold of genus g > 7 with p(X) = 1 such that its ,9roup of biregular u~utomorphisms Aut(X) ‘1s in b ‘nz ‘t e (.%. e. has a connected component Auto(X)). Then g = 12, und X is one of the following (i) X = X,, MC is th,r Mu/xi- Umem,uro, threefold, an,d Aut(X) E PGLz(@), (ii) threefold X = X$ is unique up to isomorphism, und Auto(X) E @+, (iii) X = X2$ is a threefold belonging to some one-dimensional fumily, and Auto(X) !IZ @*. The idea of the proof consists in the following. The group Aut(X) acts naturally on the family of lines r(X), the Bore1 subgroup B in Aut” (X) has a fixed point on r(X) (i. e. tl lere exists a B-invariant line C). The double projection from C is equivariant under the a&ion of B. Note that, for y < 5, threefolds X are complete intersections or covers of degree two. The finiteness of the group Aut(X) IS ‘. well known in these cases. There is a result similar to that of Theorem 3.4.10 for compactifications of (c” of index 1. The results of papers (Peternell-Schneider (1988), Pet,ernell (1989)) Furushima (1990)) Fmushima (1992)) Furushima (1993)) Mukai (1992a), Prokhorov (1991)) can be summarized in the following. Theorem 4.3.12. Let X be a nonsingular projective threefold, X be a’Ic kreducible divisor such, th,at X \ D v Cs and D N -Kx.

un,d let D c Then

(i) X = X22 C p ” is a Funo threefold of index of 1 and genus 12 with p = 1, (ii) D is a hyp erplune section of X such that C := Sing(D) is a line on X, (iii) the normal bundle of C in X is of type (1, -2), (iv) (Furushima (1993)) for the d.z’u%sor D, one has only th,e following two possibilities: a) multc; D = 3, b) multc D = 2, (v) in each of the cuses (iv) u) and (iv) b), the compuctifkution (X, D) has a 4-dimensional deformution family. To construct the isomorphism X22 \ D P C”, one uses Construction 4.3.4, (viii). F or example, in case (iv) a) we have that X22 \ D Y ~~ \ D’, where D’ is a hyperplane section of de1 Pezzo threefold I”; c I?” which is singular along a line of type (- 1,l). A ccording to 3.4.10, we have X22 \ D = Y5 \ D’ E c’, see (Prokhorov (1991), Mukai (1992a), Furushirna (1992), Furushima (1993)). The proof of the fact that X is a Fano threefold of genus 12 uses some restrictions on Betti numbers of hyperplane sections of Fano threefolds, see (Peternell-Schneider (1988), Peterncll (1989), Furushirna (1990), FW rushima (1993)).

86

Chapter

$4.4. Elementary

4. Fano

R.ational

Threefolds

Maps

with

with

p = 1

Center

along

a Conic

Let X = Xzy-z C B?J ’ +’ be, as above, an anticanonically embedded Fano threefold of index 1 and genus 9 > 5 with Pit(X) = Z H, H N -Kx. We shall suppose that on X there exists a non-degenerate conic (see 4.5.10). The following statement is due to M. Reid (Reid (1980a)). However, there was a gap in his proof (see Rernark 4.4.2). this is why we shall discuss the following proposition in d&ail. Proposition 4.4.1 (Reid (1980a)). Let X = XzgP2 c iPfl, g 2 5. Let C c X be a nonsingular conic, let cr: 2 + X be the blow-up of C, let H’ := c*H, and let E := a-l(C) be th,e exceptional divisor. Then (i)

(ii)

(iii)

Th e 1’aneur _ sYJs t em 1 - KFI

= IH* - El is base point

a generically finite morph@m (5 th,e Stein factorization, cp’: X -+ X’ is a Fo,n,o th,reefold of index Gorenstein skgulnrities and K,If F is a n,on-trivial fiber of the only the following possibilities:

free an,d determines

X C I&+‘, where g’ = g - 3: with, Cp:X’ --f x. Moreover, the threefold 1 and genrus g’ = g - 3 with cnn,onical = cp’*Kxf. morphism ye’, then F r” P1, and one has 2

4

X’,

CL) a(F) is u line crossing C, b) a(F) is u conic such that a(F) # C, (g(F)UC) = p’, po(cr(F)UC) = 1. c) Only in the case when C is of type (2: -2). In this case E ‘v IFA and F is the exceptional section, of the swfuce Fq. The number of fibers o,f type (a) is at most &finite for a sujjiciently general choice of the co,nic C. The same %s true for g > 9 and any choice of this corm.

(iv)

(v)

The number of fibers conic C l;s .su&iently (b). If F is a non-trkiul is singular.

of type (b) as a t most finite for g > 7. If the chosen general, an,d g 2 7, then there are no fibers of type fiber

of th,e morphism

‘p’, then

the point

p’(F)

E X’

Remark 4.4.2 It was stated in (Reid (1980a)) that the number of fibers of type (ii) is finite for 9 > 6. The example below shows that this is false. The fact is that Step 4 on p. 29 of (Reid (198Oa)) uses incorr& arguments. We shall irnprovc these arguments below (Lernrna 4.4.9). Example. Let V = Vs c p” be a nonsingu1a.r de1 Pezzo t,hreefold of 5 (see 3.3.1, 3.4.8). Let S c V be a nonsingular surface that is cut out by a quadric, and let 7r: X --f V be a finite cover of degree two ramified S. Then X = Xra c p7 is a nonsingular Fano threefold of index 1 and 6 with p(X) = 1 (see 5.1.1).

degree on P” along genus

54.4.

Elementary

Rational

Maps

with

Center

along

a Conic

87

A general line L on X intersects 5’ transversally at two points, t,herefore it,s preimagc C = r-‘(L) is a non-degenerate conic on X. Thus, we can obtain a t,wo-dimensional family r’ of conks on X mapped onto lines on V. A one-dimensional farnily of lines Ct crossing a fixed line L gives us a onedimensional subfamily {C,} in r’ consisting of non-degenerate conks crossing a fixed conic C at two points. The proper transform of the tonics C, on X’ are the one-dirnensional fibers of t,ype b) of Proposit,ion 4.4.1: (ii). Corollary 4.4.3. If g > 5, then every conic C on X = X~~l-~ C P”+’ satisfies condition (*) of Sect. 4.1. If g > 7, then the general conic C on x = X&--2 c IP+1 satisfies conditions (*) an,d (**). If 9 2 9, then this is true for any conic. Proof. Since g 2 5 and p(X) = 1, then X does not contain two-dimensional planes a.nd is an intersection of quadrics (see 4.1.12). Therefore (C) n X = C. This gives us that the linear system / - K,-l = IH* - El = a*IH - Cl is base point free and determines a morphism (p: Z + X C I@+‘, where g’ := y ~ 3 (SW Corollary 4.1.8). Since -K? ” = (H*-E)” = 2g’-2 > 0 (by Lemma4.1.2), then dirn X = 3, and $? is generically finite. Further, by our construction, X’ is normal, and -“x = p’*H’ for some arnple Cartier divisor H’ = p*U,(l) on X’. On t,hc other hand, the restriction ‘p’: X0 4 Xb to some open subsets 2” c 2 and X(, c X’ with codim(X’ \ X,“) > 2 is an isomorphism. It follows frorn the equality -Kx; = H’lxX that -Kx, = H’, i. e. Kx, is a Cartier divisor and “2 = $*K x,. This rncans that, X’ has only canonical singularities. But the canonical singularities of index 1 are Gorenstcin, see (Clemens-KollBr-Mori (1988)). Tl us p roves the first part of Proposition 4.4.1. By Lemma following. Lemma

(iii)

E = 4. From

this it is easy to see the

(i) If th e conic C is of type (O,O), then E r” P’ x p’, md of hidegree (1,2). If the conic C is of type (1: -l), then E = IF2, and -KTIE: = s + 3f, where s is the ezceytional section, md f is a fiber of the ruled surfme E=lFz. If the conic C is of type (2, -2), then E r” IF4, und -I(FIE = s + 4f, wh,ere s is the exceptional section and f is a fiber of the ruled surface E =IF4. -It-,-

(ii)

4.1.2: we have (-1(y)’

4.4.4.

IE

is

0, di?JiSO7-

The complete linear system Ic3E(-h’x)l IS ., b ase point free and determines a morphism E + R = R4 c lF” onto a surface R of’ degree 4 in p’ which is an isornorphisrn in cases (i), (ii), and a birational morphism which contracts s ont,o a cone over a rational norrr1a.l curve of degree 4 in p4 in case (iii). The surface E is a projection of the surface R = R4 c p’ (the case E = R is also possible).

88

Chapter

4. Fano

Threefolds

with

p = 1

From this it follows immediately that in the case when C is a (2, -2)~conic, the exceptional section of the surface E E 1F4 is contracted by the morphism (p’, and in all the other cases none of the non-trivial fibers is cornained in E. Then each non-trivial fiber F of the morphism p’ satisfies the following property: dima > 0, and C # o(F) is contained in a fiber of the rat,ional map $: X - - + X C P”+l, the projection from the pla,ne of the conic c. Therefore (a(F)UC) = P”. S’mce Pit(X) = Z.H, then dirn F = dim a(F) < 1. The curves a(F) and C are components of a one-dimensional scheme X n p3 which is an int,ersection of quadrics. We get, cases (a) and (b) from (ii) of Proposition 4.4.1. So assertion (ii) of Proposition 4.4.1 is now proved. Assertion (iii) follows directly from 4.2.6. We shall prove (iv). Assume the opposite, i. e. we assume that the morphism cp’ contracts onto a curve some surface B composed of fibers of type (b). Lemma 4.4.5. Suppose that the morphism surface B onto a curue. Th,en IbO(X,

“&I‘$

-

E))

cpl: 2

5

+

X’

contracts

some

1.

Proof. Assume that dim I- KF - El > 1. Let F be a one-dimensional fiber of cp’. Then K,- . F = 0 and E F > 0, hence (-Kx - E) F < 0. Therefore B is a fixed component of the linear system 1 - Kp - El, which contradicts Pit(X) E iZ.. H. Corollary 4.4.6. Under (E) of the surface E = p(E) P”+l. In particular, dim(E)

the conditions of LeTnma 4.4.5, the linear span either coincides with F’s+’ or is a h,yperplane in > g’.

If g > 8 and ‘p’ contracts some surfa.ce B onto a curve, then by 4.4.6 we have dim(E) > 5. Then it follows from 4.4.4 that I - K,-IIE is a complete linear system. So ,!? = E is nonsingular in the cases when C is a (0,O) or (1, -1)-conic, and is a cone over a rational normal curve of degree 4 in p” in the case when C is a (2, -2)-conic. If we suppose that the surface B is composed of fibers of type (b), then we shall obtain a contradiction with the following Lemma 4.4.7. Let F c 2 be a fiber there are two cases.

of the morph,ism

cp’ of type (b). Then

(i) q(F) is a singular point of the surface E = p(E), (ii) the reCstriction PIE:: E + l? is not birational (i. e. it is generically and deg(E) < 2).

finite,

$9.4. Elementary

R.ational

Maps with Center along a Conic

x9

Indeed, if F is a fiber of p’ of type (ii), then F E = 2. That is why F is tangent to E or intersects E at two points. This gives us (i) and (ii). Let us now consider the case g = 7. It follows from 4.4.6 that dim(E) = 4 or 5. If dim(E) = 5, then we derive a contradiction, as in the case g > 8. So now we suppose that (E) = p4. The following lemrna is easy to prove by considering a general hyperplane section of the surface R’ and computing its arithmetic genus. Lemma 4.4.8. Let R’ c E?4 be a projection of a surface R c p” of degree 4 from a point x E p5, x # R. Then deg R’ = 4, and the projection is birational. Moreover, if R’ is singular alon,g a c,urue L, then L is a line in p4. Now, we finish t,he proof of Proposition 4.4.1 in the case g = 7. Since (E) = P, we have deg E = 4. Then by Lemmas 4.4.7 and 4.4.8, J!? is singular along a line L c J!? c p4. But this means that p(B) = L, i. e. (CUa(B)) = P”, so the surface n(B) is cut out on X by a linear subspace p4. This contradicts Pit(X) = Z. H. Thus, we have proved that for g > 7 the morphism cp’ ha.s at most a finite nurnber of fibers of type (b). N ow; take the conic C sufficient,ly general, (we require that 9’ should have only a finite number of non-trivial fibers, i. e. it does not contract surfaces; this is possible for g > 7, according to what is proved above and assertion (iii)). Lemma 4.4.9. Assume th,at ‘p’: 2 curves. Then, the homomorphism p: H”(X,

is of maximal

rank

0,-(-K,-))

+

-

- E))

surfaces

on

G-K,-))

or injectiue)

Proof. Assume that p is not injective. 1 - Kg - El. For the proof of the lernrrla, Ox(-Kx

does not contract

ff’(E,

(i. e. it is surjectiue

H1(i?,

X’

Then there it is enough

= H1(X,

exists a member to show tha,t

C’,-(D))

D E

= 0.

It, is obvious that D = iV + CYE, where (Y 2 0, and nrl is an irreducible divisor. Let S E 1 - KF / be a general (nonsingular) K3 surface, and let E’ := Els, M’ := Ml.7, D := D(s. The irreducible surfaces E and M are not contracted by the linear system I ~ K,-1 = ISI, so by the Bertini theorem, E’ and Arl’ are smooth irreducible curves. Frorn t,he commutat,ive diagram

0 1 c HO(&))

0 I @

= +

H&K,-))

-

H”(a,;l(-K:))

-

H’(O,-(D))

----t

0

------f

H&D))

+

0

.L ~“(G-K,-)) I 0

1 0

Chapter

4.

Fano

Thrccfolds

Tl le d’ ivisor

we get H1(O,-(D)) ? H’(Os(D)). numerically connected: Af’

with

p = 1

D’ = M’

+ aE’

E’ = (D’ ~ trE’) . E’ = (-K,-1, - (0 + l)E’) (-K,-)” E - (a + l)E’2 = 4 + 2(~ + 1) > 0.

Thus h”(Q,,) = 1, IL’(U&D’)) This proves our lemma.

= 0: and, by Serre duality,

is obviously

E’ =

I~l(O~~(D’))

= 0.

We shall complete the proof of stat,emcnt (iv) of Proposition 4.4.1. Since ‘p’ does not contract surfa.ces, then by Lemma 4.4.9 the restriction ma.p p: H”(0,-(-I(,-) 4 H”(C?E(-K~) is of maximal rank: and because h”(0~(-K,-) = g’ + 2 = g ~ 1 > 6: then p is surjective, i. e. 1 - KATIE is a complete linear system, and E = p(E) 1s a nonsingular surface (since the conic C is chosen to bc sufficiently general, we may assume that, it has type (0,O) and E z I?’ x l?l). Then the morphism p’ has no fibers of type (b) by Lernrna 4.4.7. Proposition 4.4.1 is proved. Corollary 4.4.1

tior~

number

4.4.10. we

also

of non-trivial

Theorem

Assume

have

thut

fibers.

that g 2

Th,en,

7,

in addition to the conditions und the morphkrn, p’ has

there

exists

a line

interxectinq

of Proposionly the

o, finl;te conic

C.

(Takeuchi (1989)). Let X = Xzy-2 c I?“+’ be a nonsin,embedded Funo threefold of index 1 urLd gen,us g > 5 with Pit(X) = Z. H, -Kx = H, and let C c X be a nonsin,gulur conic satisfying condition (**). Then g < 12, g # 11, and for the contruct%on p = pR of an extremal rn!~ R in diagram (d.l.l), one has the following possibilities: gulur

4.4.11

unticanorkuly

(i) g = 5, p D -

(ii)

(iii)

2s of @pe El,

4(-K,+)

g = 6, p zs” o, f of index 1 u,nd D +- 2(-K,-+) g =

Y = X = Xs c p”, and 9 contracts a conic r c Y, type El, Y = Yl() c IF7 is also a nlonjainyular Fan,0 genus 6 with Pi 7, and let C c X be a generic conic. Then according to Propositjion 4.4.1, C satisfies conditions (*), (**) of Chap. 4.1; and this is why we may apply Theorem 4.4.11 to t,hc pair (X, C). It follows from

Chapter

92

the proof of 4.4.11 that that on X there exists Now: let y = ci and (*) of Sect. 4.1, and we 4.2.1). For the morphism 4.4.1 and 4.1.12). Case Therefore canonical

4. Fano Threefolds

p= 1

‘p’ has a non-t,rivial fiber and from 4.4.10 then we get, a line which intersects C. C c X be a general conic. Then C satisfies condition can aszume that it has normal sheaf of type (0,O) (set possibilities (see @:X + X, one has the following

(I). X = X4 c lP4 is a quartic. Then 9: X --j X is a hirational morphism, singularities.

Case (II). X = Q c IP4 is a quadric. Then of degree two. It, follows from t,he equality (a(-I$) that any divisor does not contain

with

- DE)

(-Pig

3: X’ + X ia an isomorphism. and X is normal and has only

f: X’

4 Q is a finite

morphism

= 4(fl! - p)

on Q has even degree (as a surface in P”), in pa,rticular, planes. Consequently the quadric Q is nonsingular.

The surface E c X c P’ is of degree only of degree 4 (because X is normal, need a strengthening of Lernrna 4.4.8.

Q

4 or 2. Moreover, in case (I) it can be and p: X + X is birat,ional). Now, we

Lemma 4.4.14 (R.eid (1980a)). Let R = R4 c P” be the image of@ x P’ under the map determined by the linear system o,f bidegree (1, l), and let R’ = Ri c P4 be the projection of R from u point x E P5, x # R. Then for R’ there exist the following possibilities: = ! is c1line in P4, and th,en th,e point x C c R wh,ich is mupped onA0 a line e, Sing(R’) = {y} is a point, the singularity y + R’ x of the projection P5 - - + P4, R + R’ 1’zes on a tangent line to R. In this case R’ is not a locally

(i) Sing(R’)

lies

in

the

of some

plane

cowic

(ii)

It follows irnmediately from Lemma has only the following cases: (Ia) (Ib)

X = X4 c P4, deg l? = 4, X = X4 c P4, degE = singular point ~ (Ic) X = X4 c P”, degE = 4, (IIa) X = Q c P4, deg E = quadric, and Sing E = 1 is (IIb) X = Q c P4, degE = 2,

4.4.15 that

is not normal, unique

the

a-secant

complete

for the surface

center or

on

a

in&ersection.

E c X, one

(i?) = P”, and Sing l? = e is a. line in IP4, 4, (E) = P4, and the surfa.ce l? has a unique section of X, (E) = IP”, and l? is a hyperplane 4, (J!?) = lP4) J!? is cut out on Q by another a line in IP4, section of Q. (J!?) = P “, i. e. E is a hyperplane

54.4.

Elcrncntary

Rational

Maps with

Center

along a Conic

93

Suppose that there are no lines on X. Then the morphism p’ has no fibers of type (a) (see 4.4.1)! and in cases (Ia)-(I c2) we have Sing X = Sing E’ = ‘p(D) , where B is the exceptional locus of t,he morphism cp’. Let, us consider cases (Ia)-(IIb) st,ep by step. Case following Lemma a cwve transform

(Ia). The lemma. 4.4.15. P, the*rL of some

(P)

fact that

this

If under =

p’,

hyperplane

case is impossible

conditiom and

th,e section

(I)

the snrfuce

morphism onto

is a consequence

cp’ = the

mrve

E is sin,gulnr

$T contracts

the

of the

along proper

!.

Proof. It follows from Lernma 4.4.15 that (0 # p”. Let B’ = p-‘(e). Then B’ is a surface, and B’ $ E. That is why dima = 2, and the linear span (C U (T(B’)) is generated by (C) = p2 and p”, where k = dim(@. Therefore a(H) c x n P ‘+” Since Pit(X) = Z H, this is possible only if k = 3, i. e. (e) = l?’ and CJ(U’) = X n B”. Case (Ib). Here Sing(E) = Sing(X) = {y} is the only singular point. A general hypcrplane section 3 of the variety X passing through r~ is a K3 surface with only one singularity at y which is an ordinary double point. This gives us that the quartic X also has at ?J an ordinary double point. Applying the Lefschetz theorern about hyperplane sections to t,he proper transform of X urlder the blow-up 5 + p4 of y, we obtain i. e. & is cut, out, on X by some hypersurface. l? is a hyperplane se&on of X, a contradiction

that X is a factorial threefold, But degX = deg E = 4, hence to (E) = I?“.

Case (1~). We shall prove tha,t this case is impossible for a, general conic C. We have a finite birational morphism E = p1 x p’ + i? = I& c p’; this is why the quartic E is not normal and is singular along a curve e. By Lemma 4.4.15, we have that (k) = p”, and dege > 3. On t,he other hand, a general sectiou ,!? n p2 has arit,hmetic genus 3 and is singular at the points t n p”. It follows irnrncdiately from this that dege = 3, e c p’ is a rat,ional normal cubic curve: and E has no other singular points. Such surfaces were considered in (Griffiths-Harris (1978), ch.4,$6). Thus the rnorphism cp’ = 3 contracts only a surface B’ = B, the proper transform of a hyperplane section H(C) of the variety X. Assume that case (Ic) occurs for the general conic C on X. Then for C there exists a hyperplane section H(C) swept out by a one-dimensiona~l family of tonics {C,} such that #C n C, = 2 (see 4.4.1). Note that, through each point z E C passes at most a finite number of tonics from the family {Ct} ( o th erwise their proper transforms F+ c X intersect t,he same fiber ,f of the surface E = p’ x pl, and then the points cp(Ff) lie on the line p(f), i. e. (P(B) = Sing(E) is a line in p4, a contradiction with 4.4.15). Further, because 0 = (-Kg)’ B = (-K,-)” (0(-K,-) - PE), we

Chapter

94

4. Fano Threefolds

with p = 1

have B N -Kx

- B = H* - 2E. Therefore the hyperplane section H(C) has a singularity of multiplicity 2 along C. This is why for a general conic Ct from the family {Ct }, we have H(C) # H(C,,). Let, {C,,,,} be t,he family sweeping out the surface H(Ct). We shall choose a general conic C,,, from { Ct71}. Then #G nc,,, = 2, G, $f H(C), and we rnay assume that Ct,, n C = @ (otherwise either C+,,, passes t,hrough the intersection points of Ct n C, but t,herr are only a finite number of such tonics in the farnily Ct,, or #(Ct,, (C n Ct )) > 3 which is impossible, or C+,, C (C n C,) = P, which is also impossible because X is an intersection of yuadrics). Denote by r c X t,he proper transforrn of Ctu. Since Cfl, n C = 0 and C+,, c H(C), we have t,hat -Kx r = H* . r = 2, and p(r) is a conic on X. On the other hand; #Ft n r = #C, n CtTL = 2, and Ft is a fiber of the morphism 3. Therefore the curve p(r) must be singular at t,he poirn p(F,), which is impossible. This contradiction shows that for a general conic C;, the corresponding hypcrplane sect,ion H (C,) does not exist. Case (IIa). First, we note that in this case the morphism ‘p’ does not contract surfaces. Indeed; otherwise by Lemma 4.4.15 the irna,ge of t,he contracted surface R coincides with Sing(E) = &. Then we obtain a contradiction is a finite nurnber of points. Let as in Lemma 4.4.15. Therefore Sing(X’) R c Q be the ramification divisor of the degree two cover 3: X’ 4 Q. Since we have that R also has only a finite number of Sing(X’) = G^‘(Sing(R)), singular points, in particular, R is irreducible. Further, the degree two cover 3: X’ + Q splits over surface E, so R touches E along a curve R n E’. By Lernrna 4.4.14, the preimage @-l(Y) 1s an irreducible nonsingular rational curve on E = P1 x IP’. and the restriction

is a finit,e morphism ran&cation divisor. a contradict,ion.

of degree Therefore

two. Thus l touches

the line & cannot be contained R and the restriction 3/$-1(k)

in the splits,

Case (IIb). We shall prove that in this case the variety X is isomorphic to the threefold from Remark 4.4.2. First, we not,ice t,hat the surface E is a smooth hyperplane section of Q (because there exists a finite morphism P1 x Pr -7‘ E). Further, since (E) # (Q), t,li e map p from Lemma, 4.4.9 is not of maximal rank, and therefore the morphism cp’ contracts some surface B c X onto a curve !. As in Lemma 4.4.15, we can prove t,hat) (P) = p3, and aB is a hyperplane section of X. Simple computations show that B -I(,- E = H* - 2E. Compute the degree oft, deg!

= ;(-“z

. B. E) = ;(-li;)

. (-I$

- E) . E = 3,

$4.5. Elementary

R.ational

i. e. e c l? c Q is a rational normal tive diagram of birational ma.ps

Maps with cubic

Center at a Point

curve.

Consider

95

now the commuta-

(4.4.1)

where p: a 4 Q is: the blow-up of e, 8: a + Vj is the contraction of the proper transform of E (see 3.4.1). The rationa.1 maps (i/ and 7r are const,ructed in such a way that the diagram is commutative. It follows from Proposition 3.4.1 that v? is a nonsingular de1 Pezzo threefold. Let 7’: X’ + X’ be t,he Galois involution. This involution induces two natma birational involutions 7:x--+x and 7:X--+X. Further, 7*(-K%) = -KF and 7+(E) = E (because E = p*(E)), tl lerefore 7* acts trivially on Pit(x). This gives us that the act,iorl ?‘z 2 - - + J? is biregular, $ is a regular finite morphism, and ij = X/T. S’Irnl ‘1ar 1y, one can show that 7: X - - + X is a biregular involution, and 7~ X + V, is the projection onto the quotient. Thus n-: X + Vs is a finite morphism of degree two. It follows immediately from the Hurwitz formulas that the ramification divisor R c Vs is cut out by a quadric, i. e. R N -KI/,. The existence of lines on X t,hen follows from the count of dimensions: in the two-dirnensional family of lines on vs there exists a line tangent to R, and the preimage of this line on X splits into two lines. The t,heorern is proved.

54.5. Elementary

Rational

As above, WC suppose H = -Kx. In this section this section are similar to 4.4: so some proofs will be

Maps

wit,h Center

at a Point

that X = XZgp2 c P9+l, Pit(X) = H . Z and we sha.11 assume that g > 6. nlany statements from the corresponding statements from Sect. 4.3 and omitted.

Proposition 4.5.1 (Reid (1980a)). Let X = Xzrlm2 c pY+l, g > 6, and let x E X be some point not lyin,g on a line (such a point exists according to Propositaon 4.22). Let ff: X + X be th,e blow-up of 2, let H* := cr*H, and let E := nP1(z) be th,e exceptional divisor. Then (i)

the

linear

system

a generically the

Stein

X’

is

a

Gorenstein

/-

finite factorizakion

Fmo

threefold

singulurities

KFI

= lH*

morphis-m p’:

(p: 2 X

of

~ 2EI zs‘/ b use pod

+

index

and Kx

X’,

t

X

C PY’+l,

(p: X’

1 und

,free

+

X.

genus

g’

= p’* Kxf.

and

IlJheW

g’

Moreover, =

g -

determines g - 4 with the threefold

=

4 with

car~~r~ical

96

(ii) (iii)

Chapter

IC’ E Sing(X’) 2f F = p’-’ and

one

has

u) a(F) b) a(F)

(iv) (v)

* (:c')

ij

the

fiber

the following

with

of

th,e rrt,orphi,sm

of &9ree

6 such

fibers II: (i.

z is ch,osen

$,

then

F z i?l,

possibilities:

is a conic pussiny through z, is n curve of degree 4 such that mu&

point

p = 1

(x’) > 0,

is a nwmtrivinl

only

the nvrnber of non-tkial gene& choice of point

fibers

Threefolds

dim ‘p’-l

ma) = 1, a(F) is a curve Pa(4F)) = 2,

c)

4. Fano

e. y’

suj~clently

th>at

(T(F)

= 2, (a(F))

= p’,

mult,,; o(F)

= 3, (a(F))

= i?“,

of p’ is at most jinite does

not

general

cod-act and

jar divisors),

g > 8,

then

a suficientlyl there

are

no

of type (b) and (c).

Proof. The threefold X is an intersection of quadrics, a.nd :c does not lie on a line contained in X. So Tr,x n X = 2~1; (as a scheme), and, as in tlic proof of 4.4.1, the linear system 1 - Kxl = lH* - 2El = a*IH - 2rl is base point free and determines a morphisrn $Z: X 4 X c JP’l’+l, where g’ = g ~ 4. After that, the assert,ion (i) can be proved similarly to t,he proof of 4.4.1 (i). In assertion (ii) the implication + is evident,. The implication +== follows from the fact that for any curve & c p’-‘(x) we have e (-Kg) = 0. Let F = ~p’~‘(xz) be a fib er of positjive dirnension of the morphism p’. Since E=P2and(-K~)2J3=4( see 4.1.2), we have OE(-K~) = (YE(~), i. e. the surface E = y(E) is a projection of the Veronesc surfa,cc Rq c P” from a linear subvariety, not intersecting Rd. Hence it follows immediately that, @IS: E + E is a, finite morphism, and dirn E n F = 8. This is why F ha.s t,he following property: dim a(F) > 0. and CT(F) is contained in a fiber of the rational map X- - +X’ c P’+l, the project,ion from t,hr tangent space T’,x to X at, t,he point 2. Therefore (a(F) U Tx,x) = P4. We get that dim F = dimcl(F) = 1, and a(F) is a component of the one-dimensional scheme X n P4 which is an irnersection of quadrics. Further, since 0 = F. (-IT,-) = F. (H* - 2E), then deg a(F) = 2 mult7 CT(F). We have the following possibilities: (a) (CT(F)) = I?“, o(F) IS t a conic, and a(F) = X n (o(F)), (1,) (a(F)) = IID”, o(F) is an intersection of two quadrics, a(F) = X n (o(F)), and mult, a(F) = 2, (c) (o(F)) = p4; &go(F) = 6, (T(F) is a component of the intersection of three quadrics, and multr a(F) = 3: (cl) (a(F)) = p4. dega(F) = 8! a(F) is an intersection of three quadrics, and mult,,, a(F) = 4. The last case in P4 1 WcT(F) = scheme 2 = X have iu’z = UZ(

is impossible. Indeed, in this case, by the adjunction formula is a component of the U,(F) (1). On the other hand. a(F) n P+i, a one-dimensional section of X. Moreover, we also 1). But since g > 6, then deg Z = 2g - 2 > deg a(F), so Z

g4.5.

Elementary

R.ational

Maps

with

Center

at a Point

97

has another component intersecting a(F). This is impossible. Assertion (iii) is proved. Now. we shall prove (iv). Assume the opposite; then the morphism p’ contracts some surface B c 2 onto a curve. Denote D = D, = a(B). The surface B carmot be composed of fibers of type (a) (otherwise D,r is covered by tonics passing through 2; t,his contradicts 4.2.5). Assume now that B is composed of fibers of type (b). This means that through a general point n: passes a one-dirnensiona,l family of curves of degree 4 and arithmetic genus 1 with singularit,y at 2. Denote by & the base of the scheme parametrizing such curves. It is easy to see t,hat dim& > 4. Lemma 4.5.2 (Gushel (1982)). Let X = XZ!~-~ c IP!I+l, g > 5, and let Z c X be a reduced (but possibly reducible) cuwe of degree 4 such that (Z) = p’, pa(Z) = 1, and none of the components of Z are lines. Then there exists a smooth hyperplane section passing through Z. Proof. Let A = IH ~ Zl c IHI be the linear system of hyperplane sections passing through Z. It is easy to see that for any point, z E X, we have Tr,, # (Z) (otherwise, Z c TY,x n X but, because X is an intersection of quadrics, T,,xnX is a. point or union of lines). Applying these argurllents a.nd the Bertini theorem, we obtain that a general divisor H E A is nonsingular outside Z, and it can have only isoked variable singular points on Z. In particular, H is nonsingular at Sing(Z). Let 6: 2 ---f X be the blow-up of Z. Then t,he variety 2 is nonsingular outside 8-l (Sing(Z)) and, since Z = (Z) n X, t,he linear system a*A is base point, free. The rest of the proof is similar to that of 4.2.1. Consider

the incidence R c IHi x A4,

variety R = {(H,

Z)IH

E IHI, Z E Ad, H > Z}

and projections prl: R 4

IHI,

pra: R + Ad.

The projection pr, is obviously is surjective, and its fibers a.re isomorphic to projective spaces p”-“. That is why dim R = dim & + g - 3 > g + 1. On the other hand, by Lemma 4.5.2, a general hyperplane section passing through Z is a nonsingular K3 surface and therefore cannot contain a family of rational curves, i. e. a general fiber of prl is O-dimensional, dimpr, (R) > g + 1, and pr, (R) is a dense set in IHl. Therefore a general hyperplane section of the variety X contains a rational curve of degree 4. This contradicts the following result of Moishezon. Theorem 4.5.3 (Moishezon (1967)). Let V c P”” be a nonkngular threefold, let H be its general hyperpke section,, and let i: H - V be th,e natural embedding. Then i*: Pit(V) + Pit(H) is an isomorphism if and only if one of the following two conditions is satisfied: (a)

b(V)

= h(H),

98

Chapter

4. Fano Threefolds

with p = I

(b) 1L”yV) < h2.“(H) It, remains to consider the case when the surface B is composed of fibers type (c). We nerd t,wo general results. In a way similar to that of Lemma 4.4.7. one ca.n prove the following Lemma (i)

4.5.4.

Notation,

4.5.1.

If F is a n,on-trivial

fiber

singular

s*urface

of p’ of type (b) or (c), then, either p(F) is u E = p(E), or the restr%ction ~1 E: E + l? is

fiber

of cp’ o,f type

point

of the

n,ot birutional. If F is a n,on-trivial are possible:

(ii)

as in

of

(c),

then

only

the

following

causes

PIE: E + l? is birational, und C@(F) E I!? is a point of multiplicity 3, b) 91 E: E + E is firinite of degree 2, and p(F) t E is a singular point, c) PIE: E + E is finite of degree > 3.

a)

We shall use the following surface.

classical

result

about

projections

Lemma 4.5.5. Let RI c IP”, k 5 5, be a pro,jection R4 c B” from a subspace n,ot intersecting Rg. Then are possible: 0) 1) 2) 3)

of the Veronese

of the

only

Veronese

th,e folloukg

(R’) = P”, R’ = Rq, (R’) = iF4, R’ = IP2 2s a nonsingular surface, deg R’ = 4, (R’) = p”, deg R’ = 4, Sing(R’) is (I, line of double points, (R’) = p3, deg R’ = 4, Sing(R’) is a union of three (possibly Lines

intersectl;n,g

at one

point

(the

Steiner

surface cases

coinciding)

s,urfuce),

4) (R’) = p’, deg R’ = 2, R’ c IP’ is a quadric 5) (R’) = IP”.

con,e,

Frorn Theorem 3.1.10 applied to the morphism G: X’ --f x c Pg+r, we get that eit,her 9 is an isomorphism or it is a finite morphism of degree two, and then X is a variet,y of minimal degree. That is why case 5) of Lernma 4.5.5 is impossible. From Lernma 4.5.4 we get, also t,lrat, in cases 0)) 1)) 2) t,he morphism y’ has no fibers of type (c) and in case 4) it has at most one such fiber. If in case 3) of 4.5.5 the morphisrn ‘p’ contracts a surface B, then p(B) = & c Sing(E) is a line on l? = R’. But then (O(B) U Tr,x) = p”, i. e. the surface c(B) is cut out on X by a linear subspace p5, a contradiction with Pit(X) = .Z. H. Assertion (iv) of Proposition4.5.1 is proved. In a way similar to t,liat, of 4.4.9, one can prove the following Lemma surfaces

Then

Assurer to (iv),

4.5.6. (according

the

that this

the

morphism

is true

for

u suficiently

y’:

2

-

H”(E,

+

homomorphism

p: Ho@, is of muzimul

rank.

0,-(-K,-))

U&K,-))

X’ general

does snot contract pod

z e X).

$3.5. Elementary

Rational

nlaps with Center

at a Point,

99

From the last lemma we get that for g > 8 the homomorphism is surjective, the surface E = p(E) is the nonsingular Veronese surfa,ce and, according to 4.5.4, 9’ can have only fibers of type (a). Proposit,ion 4.5.1 is proved. Remark 4.5.7 (i) Under the condit,ions of Proposition 4.5.1, asslmle in addition that g > 8. the point z E X is chosen sufficient,ly general, and the rnorphism 9’ is not an isomorphism; then there exists a conic on X passing t,hrough .z‘. (ii) Under the conditions of Proposition 4.5.1, if we have also that g < 7; and there is no conic passing through point 2, a.nd cp’ is not an isomorphism, then it, follows from the proof of 4.5.1 that, for the surface E there exist only the possibilities Z), 3) or 4) from Lemma 4.5.5. Theorem 4.5.8 (Takeuchi (1989)). Let X = XZ!~-~ c p”+’ be a nonsingular anticunoniculy embedded Funo threefold of idez 1 and genus g > G with Pit(X) = Z. H, let -N *X = H, und let x E X be u pod nwt lying on a he. Assume that the point II: sutisfies condition, (**) from Sect. 4.1. Then, g < 12, g # 11, and for the con,tmction of un extremal ray cp = q?R in, diagram (4.1.1) one has only the following possibilities: und cp contracts t,h,r davzsor (4 g = 6, Y ZY ‘* of type E2, Y = X = XI6 c pl”, D N -4K,-, - E+ to a point: (ii) g = 7, y l;s of tlype El, Y = X = X 16 C p” is u nonsingulnr de1 Pezzo threefold of de,yree 5, und p contracts the divisor D N -5Kx+ -2E+ o&o a smooth curve r c Y:, c IID’ of genus 7 and degree 12, (iii) (Tregub (1985)) g = 8, y is of type El, Y = YJ c I??’ is a nons%ngular cubic, an,d p contructs the dikor D N -3Kx+ - 2E+ ondo Q smooth curve r c Ys c p4 of genrus 0 and degree 4, (iv) y = 9, cp is of type E2, Y = Yl(; c p I0 l;s a nonsingular Fano threefold of index 1 and genus 9, and cp contracts the divisor D N -Kg+ - E+ into a point, (v) g = 10, cp %s of type Dl, Y = pl, ‘p: x+ + p1 l;s a de1 Pezzo jib,rution of degree 6 with two-dirnensiorral fibers, the divisor D N (vi) g = 12, p is of type El, Y = p”, and 63 contracts --SK,-, - 4E+ onto a srnooth curve r c IID” of genus 0 and degree 6.

Proof. The case g = 6 can be considered as the case g = 5 in Theorern 4.4.11. Let g > 7. Then, similarly to 4.4.11, besides the possibilities from 4.5.8, we get also the following ones: (vii) g = 13, Y = Q C p 4 is a smooth quadric, and p is of typt‘ El, (viii) g = 11, Y = p2: and y is of type Cl, Fano t,hreefold of index 1 and genus (ix) g = 9, Y = Y14 c p!’ is a nonsingular 8, 9 contracts a divisor onto a conic (type El), (x) g = 8, Y = Q c p4 is a smooth yuadric, cp contracts a divisor ont,o a curve of genus 7 and degree 10 (type El),

100

Chapter

4. Fano Threefolds

wit,h p = 1

(xi)

g = 7: Y = Yr2 c Ps is a nonsingular Fano t,hreefold of index 1 and gemrs 7, cp is of type E2, Fano threefold of index 1 and (xii) g = 7, Y = Yic c P7 is a nonsingular genus 6. p contracts a divisor onto a conic (t,ype El). In all the cases (i)-(xii) the defect e = E” - (E+)” of t,he elementjary rational map ‘$1:X - - + Y is different from zero, this is why the exceptional locus of the morphisrn cp’ is non-empty, and it follows from 4.5.3 that for ,y > 8 there exists a conic on X. Then we can use Theorern 4.4.11, and we obtain that cases (vii) and (viii) are impossible. In case (x) there exists a line on X, so we can use Corollary 4.3.5. In this case we get /r,1,2(X) = 5. On t,he other hand, ill.“(X) = g(f) + 111.2(Q) = 7, a contradiction. Case (ix) can be elimirnned in a similar way. Here we can use the already considered case g = 8. We omit cases (xi) and (xii), see (Takeuchi (1989)). Remark 4.5.9. From the proof of the theorem it is easy to obtain that rational maps is different from 0. the defect e = E” - (E + ) ’ of elementa.ry Therefore the exceptional locus of t,he morphism p’ is non-ernpty (see 4.1.3). Theorem 4.5.10 (Shokurov (1979b), Reid (1980a)). Let X = Xzy-~ c pgfl be a nonsingula~ anticunonically embedded Fano threefold of index 1 und genus g > 6. Then there exists a conic on X. Note that Proof. z satisfies Theorem 4.5.9 that g = 6 or possibilities:

the case 9 5 5 was discussed

in 4.2.8.

Let z E X be a general point; then, according t,o 4.5.1, the point condit,ions (*) and (**) f rom Sect. 4.1, and it is possible to apply 4.5.8 to the pair (X,2). If g > 8, then it follows frorn 4.5.7 and there exists a conic on X passing through 2. Now, we assume that, ,y = 7. As in the proof of Theorem 4.4.13, we have the following

(I) g = 6, X = p’, and 3: X’ + P3 is a finite cover of degree two, (II) g = 7, X’ = X = X4 C P is a quart?: with only canonical singularities, (III) g = 7, variety X = Q c P4 is a smooth quadric, and Cp:X’ + Q is a finite cover of degree two. We suppose now that there are no tonics on X; then the exceptional locus of t,he morphism cp’ is non-empt,y and consists of a finite number of fibers of types (b) and (c). That is why Sing(X’) = Sing(E’) is a non-empty finite set (see (ii) of 4.5.1 and 4.5.4). Let as consider cases (I)- (III) step by step. Taking into account (iv) of 4.5.1 and Lemma 4.5.6, WC can assume that (E) = P”+l. Case (I). H cre. either E c P” is a Steiner surface or E = Q’ c IID” is a quadric cone (see 4.5.5). The case of a, Steiner surface can be excluded similarly to (IIa) from Theorem 4.4.11. If E = Q’ is a quadric cone? then E’ = G(E), i. e. E’ is a Cartier divisor on X’. But then E’ is a locally complete intersection in a normal variety X’ and it, cannot have isolated non-normal singular points.

54.6.

Sane

Other Rational

Maps

101

Case (II). In this case the surface l? = E’ ca,nnot be nonsingular~ and it follows from 4.5.1 that, E = E’ cannot be singular along a curve. But 4.5.5 gives us that E = E’ cannot have isolated singularit,ies. Case (III). If l? is a nonsingular surface of degree 4, then by the adjunct,ion formula, we have K, = c?,& -l), but E r” p2, a contradiction. The case when l? is singular along a line is excluded similarly t,o case (IIa) from Theorem 4.4.11. This proves the theorem.

$4.6. Some Other R.ational Maps Elementary rational maps compose a very small class among all rationa,l maps between Fano threefolds with p = 1 (cf. Sect,. 9.2). That is why it is easy to classify all elerncntary maps wit,11 centers at points and along curves of small degree. We consider here only t,he most interesting ones. We shall use the notations of Sect. 4.1. Lemma 4.6.1. Assume that on g > 5, there exist a line and a conk. csubic

Fano threefold X Then X also contak

a

=

Xzc1-2

c

a rational

IP+l, n,ormal

curue.

Proof. A general conic Co on X intersects t,he surface of lines (see Remark 4.4.2); because Pit(X) = Z. Therefore on X there exists a reducible cubic curve C = Co U Ci, where Ci is a line. Moreover, Ca and Ci intersect transversally at, exactly one point, (Cc U Ci) = p’, and p,(C~~ U Ci) = 1 (this follows from the fact that (Co U Ci) n X is an intersection of quadrics). Since C is a locally complete irnersection, then NC/X = Hom(lc/1:*,, 0~:) is a locally free sheaf of rank 2. Let r be an irreducible component of the Hilbert scheme paramet,rizing deformations C in X. Then by deformation theory and by the RiernannRoch theorem, we have dirn r 2 h”(C,

Nc;,x)

- hl(C,

NC/X)

= deg Nc:lx

+ 2(1 -p,,(C))

= 3.

On the other hand, reducible cubic curves on X compose only a t,wodimensional family. Indeed, otherwise on X there exists a two-dimensional family of tonics contained in the surface swept out by lines on X. This con tradicts Proposition 4.2.5. Lemma

4.6.2. Let X = X2g-2 c P+‘, y > 6, let C c X be and sufficiently general, for ,y = 6, rational normal Let a:2 + X be the blow-up of C, let H* = o(-Kx), an,d let be the exceptional dtiuisor. Then the linear system 1 - Kxl = (H*

for

point

y

>

7,

free and determines

a generically

finite

morph,ism

9: 2

+

an, arbitraq, cubic curue. E = g-‘(C) - El is base X

c

IID”‘+‘,

102

Chapter

where

X’

g’

4. Fano Threefolds

with p = 1

If VI: x + X’, (p: X’ + X is the Stein, factorization, then threefold of in,dex 1 and genus g’ = g - 3 with, on,ly canonical sin,gula&ies; moreover Kx = (p’*Kxl. In particular, C satisfies (*) of Sect. 4.1.

:=

g - 4.

is a Fano

Gorenstein

condition

The proof of the lemma is similar to the proofs of Propositions 4.3.1, 4.4.1, 4.5.1. The only thing that is necessary to show is that (C) n X = C. The linear span (C) = p” of C cuts out on X a. curve which is an intersection of quadrics, i. e. th(> following two cases are possible: a) (C) n X = C; b) (C) nX = C U 2 w h ere Z is a line, p,(C

U 2)

=

1, and #C

n 2 = 2.

We shall prove that case b) is impossible. In this case the curve C is in the base locus of the linear system 1 - K x - 221. This is impossible for g 2 7 since the base locus of 1 - K x - 221 consists of lines (see 4.3.4 and 4.3.1). If g = 6: then C is contained in a unique irreducible surface F E 1 - Kx - 221 (this surface is comracted by the elementary rational ma.p with center along the line 2 xc 4.3.4). If a general cubic C c X has a 2-secant, 2, then for some line 2 the surface F E I - K x - 221 contains a two-dimensional family of rational cubic curves. This is impossible. The lemma is proved. One can show that the family of rational normal cubic curves on X = Xzr1-2 c pg+’ is three-dirnensional and dense in X, and the general such curve C has normal sheaf

In the case g = 6 this family

was investigated

in (Iliev

(1992))

Proposition 4.6.3. Let X = XzcJ/--2 c Pa+1 b e a non,sin,gular andicanonicaly embedded Fano threefold of index 1 and genus g > 6, and let C c X be a rational normal cubic curve. Assume that C satisfies conditions (*) and (**) of Sect. 4.1. Then for the contraction of an, extremal ray cp = $aR in diagram (4.1.1), one has the following possibilities: (i)

(ii) (iii)

= Xl0 c IP7, cp con,tracts a divisor onto a rational normal cubic curwe (type El), g = 7, Y = P”, cp contracts a dtuisor on,to a curve r c P” of degree 9 and genus 7 (hype El), g = 8, Y = l?‘, cp is a conic bundle with the discriminant curve of degree g = 6, Y = X

5 (We

(iv) (v)

(vi)

Cl),

g = 9, Y = Y:, c P” is a nonsingular de1 Pezzo threefold of degree 5, cp contracts a divisor onto a curue of degree 9 and genus 3 (type El), g = 10, Y = Yi8 c IlDri is a nonsingular Fano threefold of index 1 and genus 10, arrd p contracts a divisor onto a rational normal cubic curve @we El) g = 12, Y = p2, QGis a conic 3 (type

Cl).

bundle

with

the discriminant

curve

of degree

54.6. Some Other

Rational

Maps

103

Theorem 4.3.7 and Remark 4.3.9 make it possible to construct, all Fano thrccfolds of genus > 9. The following statement, gives an opportunity t,o obtain Fano threefolds of genus 7 by using the construction inverse to 4.6.3 (ii). Theorem Assume

(i)

4.6.4.

thut

The

the

lineur

(ii)

und

Ewq

it

(r,

c

P3)

system

111: IF’” - - +X 1 and ~enu.s rJ

Let r

pair

= 7.

X12

This

is inverse

Funo

threefold

I?”

be a smooth

sutisfies I01?3(15)

conditions ~

4rl

CWIJP

of

(*); determIkes

P8 onto a nonsingular m,ap is un elementary rational c

to th,e map

from

carr. be obtakled

degree

(**)

9

mu!

u

rutionul

Funo

7.

genus

of Sect. 4.1.

Then, mup

threefold of mup with center

index along

Proposikm 4.6.3, (ii). by using th,is corl,s~~lLc~ion.

(i) According to (Mori (1984)), there exists Rernurks. p” of degree 9 and genus 7 lying on a smooth quart,ic. Such intersection of quart,@ and this is why it satisfies conditions Sect. 4.1. Therefore Fano threefolds of genus 7 do exist,.

a curve r C a curve is an (*), (**) of

(ii) In the works (Iskovskikh (1979a). Iskovskikh (1988)), the existjencc of X12 was proved by using other argurncnts. In the notation of Proposition 4.3.1, the threefold x = xs c p” is an intersection of three quadrics containing a is a finite set, of points lying on E. From surface of degree 3; rnoreover Sing(X) the threefold X we can construct a nonsingular threefold 2 by blowingup t,he Weil divisor E’ c X. Cont,racting the proper transform of E, we obtain the threefold X = X12 c p8. Further, we sha.ll give another description of Farm threefolds of genus 8, different frorn 4.5.8 (iii). One more description of such Farm thrrefolds (as sections of the Grassmarmian Gr(2,6)) will be given in 5.2.3). Lemma 4.6.5 (Iskovskikh (1979b)). On the Funo p” there exists a smooth elliptic curve of de,qree 5.

threefold

X

= X14

c

Proof. We sha,ll use t,he notat,ion of-Proposition 4.5.1. The morphism (P: 2 ---f X is hirational (otherwise p: X --i X is a finit,e cover of degree two: which contradicts 4.1.10 and 4.5.4). Tl us is why X = xcj c p’ is a Fano threefold of index 1 and genus 4 with canonical singularities, X = X, and X is the intersection of a quadric Q and a cubic V in P”. The surface E is the Veronese surface in IY’“. The quadric Q is srnoot,h. Therefore there exists a smooth plane cubic curve 2 c Q n V intersecting E at one point. The proper transform of 2 on X is the desired curve. Theorem

(Iskovskikh (1979b)). (i) Let C c X = X14 c p” be u of degree 5. Thesrr C sutisjies conditions (*) und (**) of Sect. 4.1. The correspondkq elementury m,ap with center along C is birutl;onul. The mo,rphism p from (4.1.1) wntructs a divisor on,to a curve r (i. e. is of type El) a,n,d maps x+ onto a smooth cubic Y = Ys c p’. The curve r is of degree 5 and gen,us 1.

.sm,ooth

4.6.6

elliptic

curve

104

Chapter

(ii)

Let

r

5. Fano Varieties

of Coin&x3

with

p= I

Y = Ys C P” be a smooth elliptic curve of degree 5 lying cubic. Then r satisfies conditions (*), (**) of Sect. 4.1. The linear system I&(7) - 4rl d e t ermines an elementary rutional map I/I,: Y - - + X14 c P” with center along r. Th,e m,up $ is inverse to the map from, (i). on

c

a smooth

Now,

we summarize

Theorem genus

g with

(9

25

(ii) (iii) (iv)

there

4.6.7. p(X)

=

t,he main

of this chapter.

Let X be a n,onsin,gulnr 1. Th,en

9 5 12, 9 # 11, ezist li,nes and

und all the indicated tonics

X is one o,f the threefolds if 9 = 7, 9, 10 or 12, then

In the next section folds of genus g > 6.

results

we shall

Funo

threefold

possibilities

of index

1 and

for g can be reulized,

on X,

listed in Table X

12.2,

is rational.

describe

another

way to construct

Fano three-

Remurk. At present it is not known whether a variety Xl0 c lP7 is birationally isomorphic to another Fano threefold with p = 1. All the element,ary rational maps (with center at, a point, along a line, a conic, a ra,tional cubic curve or an elliptic curve of degree 4 (see 4.3.4, 4.4.11, 4.5.4, 4.6.3) give us again Fano thrcefolds of index 1 and genus 6.

Chapter 5 Fano Varieties of Coindex 3 with The Vector Bundle Method 55.1. Fano Threefolds

of Genus

p =

6 and 8: Gushel’s

1:

Approach

In Chap. 4 we mentioned that at the moment it is urlknown if t,here cxists a birational isomorphism between some Fano threefold Xl” c P7 and a Fano threefold with different numerical invariants. To study Fano thrcefolds of genus 6, one can use other met’hods: for example, construct some vector bundle and investigate the corresponding maps to Grassrnannians. This method wa,s used by the classics of Italian algebraic geometry and in a modern version was applied for the first time in (Gushel (1982))) see also (Gushel (1983), Gushel (1992)). Theorem 5.1.1 (Gushel embedded Fano threefold

cnlly is one

of the following:

(1982)). Let X = Xl0 C IP7 be an unticnn,oniof index 1 and genus 6 with p(X) = 1. Then

X

35.1. Fano Threefolds (i) (ii)

of Genus 6 and 8: Gushel’s

Approach

105

a section of the Grussmannian Gr(2,5) embedded by Pliicker I?(A”@) = P” by a subspace of codimension 2 und a quudric, the section by a quadric of a con,e W = Ws c p7 over a non,sin,gular Peszo threefold V = v, c p” of degree 5.

Threefolds Remark

of hypes (i) md (Gushel

(1982)).

(ii)

into

del

(Lrt: not isomorphic.

It is easy to see that cases (i) and (ii) have a urlifi-

cation: every Xl,) c p7 is a section of Gz) c pl”, a cone over Gr(2,5) by a subspace of codimension 3 and a quadric. This gives us immediately threefolds Xl0 c p7 of type (ii) are specializations of threefolds of type

p” t,hat (i). c

Proof. Simple computat,ions show that thrcefolds from (i) and (ii) are Fano threefolds of index 1 and genus 6 with p = 1. We shall prove that threefolds of t,ypes (i) and ( ii ) cannot be isomorphic to each other. Assume that some threefold X = Xl0 c p7 of type (i) is isomorphic to a threefold of type (ii). Then X adrnits an embedding i: X c-) Gr(2,5) = G such that 0x(-Kx) = i*oc(l). On the other hand, Xl0 c p7 is a section of the cone IV, over Vs c P”. According to Theorem 3.3.1, we can assume that V.5 = Gr(2,5) n P”. We shall use the standard notation for Schubert cycles, see for example (GriffithsHarris (1978)). Let P2 c P” c p4 be a flag in p4. Consider the corresponding Schubert subvarieties ~1 = al,~~(~‘) and g2 = a2,o(P”) = Gr(2,4) in Gr(2,5). Then o1 > CJ~. It is known that, CT~ is singular along a two-dirnensional plane ~~2.2 = cr2.2(P0:‘) c Gr(2.5). F or a sufficiently general choice of t)he flag p2 c p’ c p*, the int,ersection T := ‘~1 n i(X) will be a nonsingular hyperplane section, and 2 := p2 n i(X) c T will be a smooth ellipt,ic curve of degree 4. Consider the projection from the vertex wg of the cone IVS. Then the curve 2 can be mapped onto a conic or again onto a, srnooth elliptic curve of degree 4. But the threefold Vs does not contain elliptic curves of degree 4, this is why the image of 2 is a conic. Therefore 2 lies on a quadric cone K c lV5 with vertex at) wg. We get wg E (K) = (2) c (T) c (01)~ i. e. the point UIO lies in the linear span of all special Schubert subvarieties (~1. This is impossible because the locus {gl} spans IED”. Lemma 5.1.2 (Gushel gree 4 on Xl” C IF7.

(1982)).

Th ere exists a sm,ooth elliptic

curue of de-

Proof. Note; first, that if on X there exists a reducible curve 2’ = C U C’, where C and C’ are tonics, (2’) = p’, and p, (2’) = 1, then on X there also exists a smooth curve of dcgretc 4 and genus 1. Indeed, by Lemma 4.5.2 there exists a smooth hyperplanr section passing through Z’, a K3 surface H. Then 2’ varies in an cllipt,ic pencil in H. Further, we prove the existence of a pair of tonics C, C’ on X such that, (C U C’) = IF” and p, (C U C’) = 1, i.e #C n C’ = 2. Assume that thtrc is no such pair of tonics on X. We shall use Proposition 4.4.1. Then the morphism y’: 2 + X’ has no fibers of type

Chapter

106

5. Fano Varieties

of CoirrdrxS

wit11 p = 1

(b); and for a sufficient gcncral choice of conic C it has only a finite number of fibers of type (a). For the variety X C P”, one has the following t,wo cases: (I)

X = X, c PA IS a quartic with isolated tcrrninal singularities, and G: X’ + X is an isomorphism~ and $9 X’ + Q is a finite cover of X = Q c r?” is a smooth yuadric. degree two.

(II)

In case (I) the surface E = E(E) c IID has degree 4, so it is singular (see 4.4.14). Therefore t,here exists a fiber F of t,he morphism ‘p’ such t,hat, F. E = 2. Such a fiber is of type (b). In case (II) the surface l? cannot hc of degree 2. Otherwise E’ = G*(E) is a Cartier divisor, X’ is a factorial variety, and Pic(X’) = z, so 9’ cont,racts some divisor. Therefore dcg E = 4, and E is cut out on Q by another quadric Q’ c p4. Then E = Q n Q’ is singular along some line (see 4.4.15). Wc get a contradict,ion as in 4.4. This proves the lemma. Lemma

(Gushel de,qree 4, th,rough 2. Then E of ran& 2 on X

elliptic

5.1.3

C~UTW

pussin,g

bundle

th,e follo~wi~ng

h”(E)

of

Chern

(1982)). Let 2 c X = Xl0 c P7 be a smooth and let H br: a, sm,ooth hyper$une spctiorr of X to the pair (2, IT) or~e cm associate some wctor which is gen,ern,ted bg its global sections md has classe~s and cohorrrology: cl (E) = H 1 c2 (E) = Z und

= 5.

Outline of proof. Let, p: P’ x X + X br the trivial p’-buridle on X: and let pH: p’ X H + H be its restriction t,o H. Choost a srnoot,h section Y C p1 x H, Y - q) x H + p1 x Z, where .q) E p’ is a point, and 2 c H is t,hr elliptic curve from the hypothesis of the lemma. Let D - q) x X +@ x H be a divisor on p1 x X: and let J4 be the idcal sheaf of Y in pr x 11 under the natural embedding Y c p1 x H c IP’ x X. Set E := p,(C$,,(D)) @ J4. It is ea.sy to see that E is a locally free sheaf of rank 2. The linear system /Z 1 on the K3 surface H is a pencil of cllipt,ic crrrves without base point,s and fixed components. One has the following exact SCqucnce: 0 Dualizing 0 +

+

OH-Z)

this sequence 0,(H

and tcnsoring

~ 2) +

Denote V := H”(~H(Z))*. an embedding Y c IID’

H”(UH(Z))

x

H”(C?H(Z))*

@ 01, +

U,(Z)

it by Uf$(H), 8 01{(H)

+

0

we get %

OH(H

It IS ” c21ear that the homomorphism II. WC have the following c:ommut,ative

+ Z) +

0

(2 d&ermines diagrarn:

$5.1. Fano Threefolds

of Genus 6 and 8: Grrshel’s

0 0

T + Uff(H-Z)

0

-i

vxux T

--j

T muff(H)

+

uH(H+q

+

0

---f

1‘ VceUx(ff)

4

uH(H+Z)

--f

0

T V~UX T

e

0

0

gives the following 0 +

107

0

T E 1‘

The left colunrn

Approach

v -

H”(E)

exact z

sequence

H”(Ujy(H

of cohomology: - 2))

-

0

The surjectivit,y of the homomorphism r’; follows from the fact, tha.t H1 (Ox) = 0. Let us consider the following cornmutative diagra.rn with exact rows and columns: 0 L Ker y

0

+

II V@Ux

+

L E5

0 1 Ker 6

i

Lh QFf(H - Z)

+

0

It is clear from this diagram t,lrnt, the horrromorphisn~ y is surjectivr if a,rrd only if the homomorphism 6 is surjective, i. e. E is generated by globa. sections if and only if Uf[(H ~ Z) ..1s g encrated by global sections. But the propert,y of UH(H ~ Z) to be gcncrat,ed by global sections follows from the facts that, Z = Xn (Z) = XnP”, and tht the linear system IH-Zl cuts out on Ei a complet)e linear syst,em 1(H-Z) I, / wi tl rout base points and fixed components. Therefore E is grnerated by global sections. The Chern classes of E can be computed from the lower exact sequence of the previous diagram. Since ~“(UH (Z)) = 1 and hO(U~(H - Z)) = 33. then /I”(E) = 5. This complet,es the proof of Lemma 5.1.3. Now, we cornplct,e t,he proof of Theorem 5.1.1. Thr vector bundle E determines a morphism ‘41~= $1~: X + Gr(2, 5) of X to the Grassmannian such that /J*(E*) = E, where I is the universal vector bundle on Gr(2.5) of rank 2 frorn the exact, t,riple

Let i: Gr(2,5) -+ p” be t,he Pliicker embedding. Set 9 := i o SW. Since q(E*) = i*Ux(l): t,hen H = q(E) = ~p*U~;~(~,~,(l). Therefore the map p is determined by some subsystern of t,he linea,r system IHI on X. Since H” > 0: then dim y(X) = 3, and H” = 10 = tlcgpdcgp(X). On the other

108

Chapter

5. Fano

Varieties

of CoirdexS

with

p = 1

hand, because 4 = H. 2 = cl(E) . (Q(E) = p*cl(l*) . p*(E*), we llave that deg cp divides 4. Therefore either deg p = 1 and deg p(X) = 10 or deg cp = 2 and degcp(X) = 5. We get cases (i) and (ii). This proves Theorem 5.1.1. A similar

method

can be applied

to study

Fano threefolds

of genus 8.

Theorem 5.1.4 (Gushel (1983), Gushel (1992)). Let X = X14 c p” be an anticanonically embedded Fano threefold of index 1 und genus 8 with p(X) = 1. Then X is a section of th,e Grassmannian Gr(2,6) embedded by Plikker into n(r\‘C”) = i’?l” by a subspace of codim,ension 5. According to Lemma 4.6.5, there exists a smooth elliptic curve 2 of degree 5. It is easy to show that the curve 2 can be chosen on a smooth hypcrplane section H. Then, as in Lemma 5.1.3, one can prove that ihe pair (2, H) determines a vector bundle E of rank 2 such that 1) E is generated by global 2) q(E) = H, Q(E) = 2.

sections,

and ho(E)

= 6,

As in the proof of Theorem 5.1.1, the vector bundle E deterrnines a morphism 4) = dE:X --j Gr(2,6) such that E = $*(E*)! where I is the urliversal bundle on Gr(2,6). Take the cornposition of $ with the Plucker embedding i: Gr(2,6) ‘---f @* cp=io?/i:X+Gr(2,6)qpi4. Then dimp(X) assertion.

= 3, degcp = 1 and deg’p(X)

= 14. This

gives us the desired

$5.2. A Review of Mukai’s Results on the Classification of Fano Manifolds of Coindex 3 Throughout, the rest of this chapter we shall denote by X a nonsingular n-dimensional (n > 3) Fano manifold of index n - 2 (i. e. of coindex 3), and by H the fundamental divisor on X. Theorem 5.2.1. Let X be an n-dimension,al Fano manifold of index ‘r-2, and let H N -&h7x be a fundamentu.1 divisor on X. Then for yenerd HI n . . n HTL--:! is a nonsingular divisors HI, . , HT1-2 E 1H 1, th,e intersection irreducible K3 surface. Note, that for n = 3 this Theorem is Shokurov’s result (see 2.3.1). For n = 4 it follows from (Wilson (1987b)> Prokhorov (1994a)), see (Della. (1996), Mella (1997)) for arbitrary n. We shall also suppose that p(X) = 1, i. e. Pit(X) = Z. H. In Chap. 7 we shall study Fano manifolds with p > 2. In particular, it will be shown that for dimX > 6 the condition p(X) = 1 is satisfied automatically. According

55.2.

Mukai’s

Results

on the

Classification

of Fano

Manifolds

109

to 2.4.7, the condition p(X) = 1 gives us that t,he fundamental linear system IHI determines a morphism qHi:X 4 Pg+rr-z, and for y > 4 the map ‘~1~1 is an embedding. We consider mainly the case g > 6 (for the case g < 5, see 2.1.17: 2.1.5 (iv)). Mukai’s approach to the problern of the cla,ssification of Fano manifolds, as well as Gushel’s approach, is based on the construction of some vector bundle E on a. manifold X that, deterrnines an embedding of X into a Grassma.nnian. The results arc taken from (Mukai (1988), Mukai (1989), Mukai (1992a)). First, we consider some examples. Examples

5.22.

(i)

g = 6. Let Cy,, c IP1” be a smooth

Gr(2,5) c P”” over the Grassmannian Gr(2,5) c IP(r\“C’“) Then CF,, c IPIO is a fundarnent,al rnodel of a B-dimensional index 4 and genus 6 with p = 1.

sect)ion of a cone = IP” by a quadric. Fano rnanifold of

(ii) g = 8. The Grassmannian Cf, := Gr(2,6) c IP(r\“@“) = IP14 is a fundarnent,al rnodel of an 8-dimensional Fano manifold of index 6 and genus 8 with p= 1. (iii) g = 9. Let V = @” be a 6-dimensional vector space with non-degenerate skew-symmetric bilinear form C. All 3-dimensional subspaces W c V such that c(W, W) = 0 forrn a 6-dimensional subvariety Cf, c Gr(3, V) c PI’, the fundamental model of a Fano manifold of index 4 and genus 9 with p = 1.

(iv) g = 4-linear F(W, W, P(A”V) genus 10

10. L et V = C7, and let F be a non-degenerate skew-symmetric form on V. All 5-dimensional subspaces W c V such that W, W) = 0 form a 5-dirnensional subvariety Cf, C Gr(5, V) C = P, the fundamental model of a Fano manifold of index 3 and with p = 1.

(v) g = 7. Let V = C”, and let q be a non-degenerate symmetric bilinear form on V. All 4-dirnensional subspaces W c V such that q(W, IV) = 0 forrn a lo-dimensional subvariety C$’ in Gr(4, V), a Fano manifold of index 8 and genus 7 with p = 1. The fundamental model C:t c IP1” is deterrnined by the embedding by spinor coordinates. (vi) y = 12. Let V = C7, and let 01, cr2, (T:< be a triple of skew-symmetric bilinear forms on V (we assume t,hat 01, g2, 03 are linearly independent and satisfy a certain open condition). Let X = Gr(3, V, 01, ~2, ~3) c Gr(3, V) c p(r\“V) be the subvariety consisting of three-dimensional subspaces W c V is a such that al(PV>w) = az(W, W) = o:~(W, W) = 0. Then X = Xi, fundamental model of a Fano threefold of index 1 and genus 12 with p = 1. Remarks. ples above are groups: E.$sz 7, 8, 9 and 10,

(i) For 7 < g < homogeneous spaces = G/P, where G = respectively, and P

10, the manifolds E$c2 from the examunder the a.ction of sernisimple algebraic SOLO, SLe(@), Sp,(@) and G2 in cases c G is a rnaximal parabolic subgroup. It

110

Chapter

follows 2.1.9.

5. Fano

Varieties

from this that, p = 1. F’or the computation

on GG). a subspace

For g = 12 t,he threefold (al, g2,03) C r\‘V*.

Theorem

5.2.3

he a n-dim,en,sionul

p = I

of Fano indices

X,, ’

- Gr(3,

V, ~7~) 02,~)

see

is defined

(1989), Mukai (1992a), Mukai (1992b)). Fun,o munifold of index n ~ 2 a,nd genus thut X satisjies condition (**). Then, 3)

(i) 9 5 12. 9 # 11, (ii) $6 < g < 10, th, en X is a linear section of some “ma~imul” man,$old ~~~~J2 c P.~+~L(!I!F~from Exumples 5.2.2, (i)-(v), (iii)

of G’/P:

(nlukai (n, 2

= 1. Assume

p(X)

with

arc unique up to isomorphism. For .Z$!\ by ‘a quadric & c P1” which cuts out CT{,

(ii) For 7 < g < 10 the manifolds g = 6 the manifold CyO is defined

with

of Cointiex3

2f g = 12: tl hen n = 3, und X is isomorphic 5.2.2, (vi).

to some threefold

by

Let X g 2

6

n-dimensional from

Exurn-

ple

We give a sketch of the proof of Theorem 5.2.3. First, WC give a simple proof of the boundedness of the genus following (Mukai (1992a)). In the proof of this fact we can assume that dimX = 3. Let T, be the moduli space of polarized K3 surfaces (S, h) of degree 2g - 2 in P, where y > 6. A general hyperplane section h of such a surface S is a canonical curve in P”-‘. This gives us the rational map & from the P-bundle

over Fg to t,he moduli Lemma

x+2 na,lI mup

c

rP

space M,

of curves

of genus 9.

5.2.4 (hlukai (1992a)). If there exists a, Funo threefold X = +’ of index 1 nnd genw g > 6 with p(X) = 1, then the ratio-

$y: PTI - - + M,

is not

generically

finite.

Proof. Take a general pencil P c (H( of hyperplanc sections of Xzy--2 c P+’ Since P contains a singular surface, thr> image of the correspdnding rational map P-F,] is not a point. Each surface S E P contains a smooth canonical curve C; t,he base locus of the pencil P. This is why t,he rational map 49: Pg - - + M, is not finite over t,hc point, [C] E M,. Simple argurnents from deformation theory show that the point [C] is general in the image of ol. i. e. f& is not generically finite. The statement, about, the boundedness p = 1 is a consequence of the following Theorem genericully

of the genus of Fano manifolds

5.2.5 (Mukai (1992a)). The rutionul finite if and onlyl i,f g = 11 or g > 13.

mup

qbII: p:, ~ - + M,

with

is

35.2. Mukai’s

Results on the Classification

Since dim z), = 19 + g and dirnM, = dim Pg > dim M,, and the ma.p @y cannot Mow we give a sketch of the proof 411: PII - - + M11. Let C c P5 be a smooth be a smooth intersection of three quadrics be a general hyperplane st&on of S. Then is a stable curve of genus 11. Proposition 5.2.6 (Mori-Mukai of the curve r into a non,singular I: S 1 S’ such that the restriction

Ilr

111

3g - 3, we have for y < 10 that be finite. of the finiteness of the map elliptic curve of degree 6. Let S containing C, and let H := S n P4 S is a K3 surface, a,nd r := H U C

(198313)). K3

of Fano Nlanifolds

For any embedding

surface,

coincides

there

with

ezists

i: r q

S’

an, isomorphism

i.

The last proposition gives us that the point < E P11 corresponding to (S, r) is isolated in 4;: (~11 (c)). Therefore 4 11 is finite at the general point. Remark @ukai (1992a), Mukai (1988)). Tl le map & is of maximal rank at the general point except for the cases g = 10 and g = 12. In the case g = 10 the irnage of 4!I is a divisor on M 10. Now,

let X = Xzy--2 c IPY+’ be a Fano threefold of index 1 and gemIs g, with p(X) = 1. Take a general section S = XnP which (6 5 Y 5 12, g # 111, is a nonsingular K3 surface. By Moishezon’s theorem (see 4.5.3), p(S) = 1. Further, we construct, a rigid vector bundle E’ on S. Invariants of E’ are given in the table below

X(E’) 5 10 6 6 7 7 Using the following to a vector bundle

Theorem E on X.

of Flljita,

det(E’) OS(l)

c2(E’) 4

24 5 8 12 10 the vector

C3s(2) OS(l) 0s (1) OS(l) OS(l) bundle

E’ can be ext,ended

Theorem (Fujita (1981)). Let A be an ample divisor on a nonsingular iW with dimM 2 3, and let E’ be a vector bundle on A such that H2(A,End(E’) @CC3A(-tA)) = 0, Y’t > 0, and H”(A,E’ @ 13A(tA)) = 0, Vt E Z, 0 < p < dim A. Then there exists a vector bundle E on, M such th,at EIA = E’. variety

Denote V := H”(X, E) and r := rkE. Then from the table above we get that dimV = 5, 10, 6, 6, 7 and 7 in the cases g = 6; 7, 8, 9, 10 and 12, respectively. As in 5.1, the vector bundle E on X determines a morphism to the Grassmannian @ = @peel: X --j Gr(r, V) such that, E = PE, where E is the universal bundle on Gr(r, V). Further, for g = 6, 8, 9, 10 and 12 we consider the hornornorphism

112

Chapter

5. Fano Varieties

A,: A~V Y H”(Gr(r,

of Coindex3

V), A’E)

+

with p = 1

HO(X,

ALE)

for 0 < i < r. The homomorphism X, is irljcctivc for % 5 max{r 2, l}. If X, is not injective, t,hen its kernel I, := Ker X, defines a globa. section s, t HO(Gr(r,V),Ai& @Z1:) = H”(Gr(r,V), (/?\sE)t”iirrllt), and t,he scheme Gr(r, V, IL) o f Teroes , of t,he section s, cont,ains the image G(X). In the cases 9 = 8, 9, 10 and 12, the kernel I,- 1 is of dimension 0, 1, 1 and 3, respectively. Theorem 5.2.3 is a consequence of the following Proposition Gr(r, V, IT-i) Furthermore, a linear

(ii)

(iii)

5.2.7

(Mukai

?,s an the

sectio,n

morphism of

(1992a)).

n(g)-d~rnensional

Gr(r,

@: X

+

(i) Let g = 8, 9, 10 or 12. Then Fano manifold of index n(y) - 2. Gr(r; V, I,.-1) 1s ’ an ~so,rrrorph%srrL onto

V, 1,.-l).

Let g = 6. Then, the cokernel of the hornomorph%sm X2: r\“V + H”(X, A”E) is of d’vrnension ut most one. If X2 l;s surjective, then @ is an komorphism onto u complete intersection of Gr(r, V) with a lineur subspace md a guadric. If X2 is snot surjecti,oe, then @ is a finite m,orph,ism of degree 2 onto a smooth linear section of Gr(r, V). of the natural homomorphism S”V + Let g = 7. Then the kernel H’(X, S2E) is a one-dimensional vector space generated by u nondegenerate symmetric ten,sor 0. The pair (E,o) induces an isomorphism, between X and a lznear sectio,n of the IO-dimensional spinor subvariety in P’” connected ‘with the pair (H”(X, E), c).

Theorem 5.2.3 shows that Fano manifolds of genus 12 play a very special role in the classifications of Fa.no manifolds of coindex 3: their dimension does not exceed 3: and any such variety X22 cannot be an ample divisor on a nonsingular fourfold. Following (Mukai (1992a.)), we shall give other is connected realizations of the threefolds X 22 c p13. The first realization with the cbssical projective geometry of plane curves a.nd polar rb-gons. Let C c I!@ be a plant curve of degree c& and let a(~:, y, Z) = 0 be the equation defining C. A polar n>-gon of C is t,he union of n lines !, C p2 with equations fL(5, y, +z) = 0 satisfying the condition F~(z; y, Z) = C a,,fF for some ai E cc. Natural questions arising here are 1) Is the variety V(F,I, n) = V(C, d) of all polar C C p2 non-empty? 2) What is the dimension of V(C, d)?

n-gons

of a fixed

curve

(we consider V(F,I, n) as a subvariety in S”p2*). It, is known for example that for a general form F~(z, y, z), the variety V(Fs, 7) consists of one point (Palatini (1903)). Polar triangles of a smooth conic C c p” have a. very simple structure. Exumple 5.2.8. Let C c p” be a smooth conic, and let e,, lx, Ls be different lines on p*. Then it is easy to see that the following conditions arc equivalent: (i)

the t,rianglr

n := e1 + !J2 + e:, is a polar

3-gon

of C,

55.2. Mukai’s (ii)

Results on the Classification

of Fano Manifolds

113

wit,h respect to the symmetric bilinear form the triangle A is self-polar q( : ) corresponding to C (i. e. the vertices of A are orthogonal to t,he opposite sides).

Denote by v(Fd, nj) a compactification take v(P’ 2. If there exist ample divisors H17 , HI,-1 such th,nt Kx . HI . . H,-l < 0, then X is uniruled. In particular all Funo varieties are uniruled. Corollary 6.1.6 (Kawarnata-Matsuda-Matsuki (1987)). jective variety with only Q-factorial log term&al singulur&ies characteristic char k = 0. If on X there exists a nurnericully ray, then X is uniruled.

Let X be a prooiler a field k of eflective extremal

Corollary 6.1.7 (Kawamata-Matsuda-Matsuki (1987)). Let X be a projective variety wG% only Q-factorial log terminal singularities over a field k of characteristic char k = 0. Assume that on X there exists an extremul ray R of diuisorial type. Then the exception& divisor of R is a uniruled variety. Not,e that, in the case char k = 0 the uniruledness of a. variety X irnplies K(X) = -co. The Llinirnal Model Program and the Alxmdance Conject,ure in dimension 3 gives us an inverse result. Theorem charucteristic

6.1.8. Let X be a smooth prqjective three,fold over CLjield char k = 0. Then the following conditions are equivalent:

k of

(i) X is uniruled, (ii) /5(X) = --x. Now, we give a sketch of the proof of Theorem 6.1.4 following consequence Mori (1986)). Implication (i) + ( ii ) 1s an irnrnediate lary 6.1.3, so we shall prove (ii) + (i).

(Miyaokaof Corol-

$6.1. Unirulrdrless

119

Proposition 6.1.9 (Miyaoka-Mori (1986)). Let C be a, smooth p’r~@:t%~7~c: ~IL~II~:, let X be a nonsingular project& varl;eQ~ of dimension n > 2 over k, char k > 0. Let ,f: C + X be a morphism such that f(C) is n,either n point nor n mtiond CWW, let H be u’n ample divisor on, X, cd let r be a n,atural number. Assume that for every closed subscheme B c C of degree r the sch,eme Horn(C,X;flu) is ofr, osi t'uue d.zmt~n,sion, , at the point [f]. Then there exists a (possibly., sin g IL 1ar ) ru t;lon,nl curve L on X pussing throsugh a gen,eral pod of f(C)

such

that

H. The proof Let us consider [f], and let A F: Horn XC + base locus, we

L < 2/r drg f *H.

is based on Mori’s bend-and-break technique (Tori (1979)). a smoot,h curve A’ c Hom(C, X; fly) cont,aining the point > A’ b e a smooth compactification. The universal morphism X induces a rational map g: A x G ~ + X. By resolving the get the diagram below

where ,LL is a composition of blow-ups, and h maps a nonsingular surface S onto a surface in X (because r 2 1 and C is not, rat,ional). Easy computations in the group Pit(S) shows t,hat the curve L can be obtained as the image of some exceptional curve of the morphism p. Moreover, such a curve L contains one of the points of f(B). By a small deformation of the subscheme B c C, WC get the required assertion. Now, the implication (ii) += ( i ) 1s a consequence of the following. Proposition nonsingular

6.1.10 projective

(Miyaokti-Tori variety

(1986)).

of dimension

on X, let C be a closed curve on, X and Kx C < 0, then there exists a ( possibly with

the

following

(i) L 3 x, (ii) H. L < max{H

Let char k > 0. Let X be a Let H be an ample divisor let x E C be a general point. If singular) rational curzle L on X

‘n, > 2.

properties

C, 2nH.

C/(-Kx

. C)}

The proof of this proposition consists of three a,rguments for the proof of Hartshorne’s conjecture

steps. Tori used (Mori (1979)).

similar

Step 1. Existence of deformations, char k = p > 0. Let Y: C’ + C be t,he normalization of C, and let F: C” + C’ bc the Frobenius morphism of y in sucli a. way t,hat degree 4 := p”‘. Denote f = 71 o F: C” 4 X. We dioose r does not exceed ;((-Kx Then have

for every closed subscheme

C)Y - l)(.Y(C’)

- 1).

B c C” of degree

r. t>y Corollary

6.1.2, we

120

Chapter dirn[fl

Hom(C”,

6. Boundedness X; fly)

and Rational

> q(-Kx deformations,

Connectednrss

C) - nr - n(g(C”)

- 1) > 0 ,

so there are sufficiently sition 6.1.9.

many

and it is possible

to apply

Propo-

Step 2. Proposition may apply Proposition

6.1.10 in chark > 0. According t,o Step 6.1.9 and obtain 6.1.10 in the case char k > 0.

1, we

Step 3. Lifting in char k = 0. Let char k = 0, and let T E C be a general closed point. We may a.ssume that X, H, z and C are defined over a finit,ely generated domain R over Z. Also, we may assume that X and C are smooth over Spec R. The family of curves C on X which satisfy the conditions of Proposition 6.1.10 forms a quasiprojcctivc subvariety V in the corresponding Hilbert scheme. According to Step 2, the fiber of V over every closed point of Spec R contains a closed point. Therefore the same is true for the fiber Vj over Speck, the general point of Sprc R. This proves Proposition 6.1.10.

56.2. Rational

Connectedness

of Fano Varieties

Now we discuss the rational connectedness of algebraic varieties and the closely related problem of the boundedness of the degree of Fano varieties. Throughout this section; if the opposite is not explicitly specified: we shall assume that all the varieties are defined over an algebraically closed field of characteristic 0. By the degree of a Fano variety X we mean the number di111(X). By a rational curve we mean an irreducible curve which is (-Kx) birationally equivalent t,o p’ . Definition-theorem 6.2.1 (Koll&-Miyaoka-Mori (1992b)). Let X be u complete algebraic variety. Then X is said to be rationally connected if one of the follofuing equivalent conditions holds: (i) for every collection of points 21, . . . , x,, E X, there exists a rational curve on X containing x1, , x,, (ii) for every pair of points x1, x2 E X there exists a rational curve on X containing 51~ x2, (iii) for a suficiently general pair of poin,ts (x1 ,x2) E X x X, there exists a rational curve on X containing x1,22. If moreover X is nonsingular, to one of the following: (iv) (v)

then conditions

(i), (ii),

(iii)

are equivulent

every pair of points x1,x2 E X, there exists u connected rational curves on X containing x1 and x2, there exists IL morphism f: IP1 + X such that the vector bundle ample (i. e. f*Tx = @Ok, ui > 0).

for

chuir~

of

f*Tx

is

56.2.

Rational

Connectedness

of Fano

Varieties

121

It is sufficient to prove Theorem for a projective nonsingular variety X. Implications (i) * (ii) * (iii) + (’iv ) are obvious. The equivalence of (iii) and (v) follows from 6.1.3. For the proof of (iv) + (i), one uses the following Gluing Lemma. Lemma 6.2.2 (Kollar-Miyaoka-Mori (1992b)). Let g: D + X be a nonconstant morphism from u smooth projective c’urue of genus p to a nonsingular com,plete variety X. Let B,, . : B, be arbitrary points on g(D). Let N >> 0 be distinct points on D \ ,9g1({B1,. . , B,,}), and let pl,...,pN, h,:P 4 x, i = l,.. . , N be non-constant morphisms such that h:Tx am non-negative bundles, each image h,(IF’l) is separated from {Bl, . , B,}, and subset I c [l, N], a h(O) = s(E) f or all i. Then there exist a non-empty flat family of CUT~ES C + (S, o) over a one-dimensional base and a morphism f: C + S x X such that a) ,for a general point s t S, the ,fiber C, is n,onsingular, non-constant family such, that f,s(C,s) 3 B1, . , B,, b) C, is a semistuble curve which is isomorphic to (MI@) points 0 E (i-th pl) and P, E D are glued for each i, c)

fo

=

ulh,

U

D,

where

is a the

u 9.

Note that rationally connected surfaces following example shows that rationally are very close to rational ones. Proposition 6.2.3 Let X be a nonsingular (i) H”(X, (ii) H”(X: (iii) 7ri(X)

and fS(C,)

are exactly connected

(Kollb-Miyaoka-Mori complete variety.

(Qk)@‘“) = 0 for e7xry n 0~) = 0 for e7iery i > 0, = 0 if k = @.

>

the rational surfaces. The varieties of any dimension

(1992h),

Campana (1991b)). connected, then

If X is rationally

0,

Let us prove for example (i). Assume that, HO(X, (Qi)@7’) # 0, and let w E H”(X,(R:,)@“). S’mce X is rationally connected, it follows from 6.2.1 (v) tha,t for a general point 2 E X there exists a rnorphisrn f: iF1 + X such that f(0) = 5 and f*(Tx) is an arnple vector bundle. Then f*((Q&)@“) is negative. Therefore f *w = 0. So w vanishes at the general point 5 E X. This gives us w = 0, i. e. P(X, (Qi)@n) = 0. It, turns out that every uniruled variety adrnits a unique structure of a “maximal fibration” onto rationally connected varieties. Theorem 6.2.4 (Kollar-Miyaoka-Mori (1992b), Campana (1992)). Let X be a nonsingular complete variety. Th,en there exist a den,se subset U c X and a proper smooth morphism f: U + Z with the following properties.. a) every fiber

off

is rationally

connected,

122

Chapter

6. Bourdcdr~ss

and Rational

b) for a sufficien,tly yer~rul point z E Z, such that D n f-‘(z) = 0 und dirn(D

there

Cormectcdrless

are

n f-‘(z))

‘no rution,ul

= 0.

Definition. A p’roper surject%ve morphism f: U + Z dition u) of Theorem 6.2.4 is called a rationally connected RC-Jbrution). This morphism f is ca,lled a maximally fibration (or simply MRC-fib&ion) if it ulso satisfies situation the variety Z is called a, rational quotien,t of X. Remark. Under the conditions if and only if dim(Z) < dim(X). only if Z is a point. Corollary 6.2.5. Under dim(Z) < 2, then the ration&

of Theorem Tl ie variety

the conditions quotient

Z

D c X

curues

which,

satisfies

fibration,

con-

(or simply

rationally

con,nected

condition

b). In this

6.2.4, the variet,y X is uniruled X is rationally connected if a,nd

of Theosrern %s not

6.2.4, if

dim(X)

-

unkxled.

Indeed, assume the opposite. Let C c Z be a general rational curve, and let Xc := ,f-l(C). If dim(X) = dim(Z)+l, then fc:: Xc -+ C is a fibration by rational curves. Then by Tsen’s theorem, fc has a section (r: C ---f Xc;. We get a rational curve o(C) c X which does not sat,isfy condition b) of Theorem 6.2.4. Similar argunients can be applied in the case dim(X) = dim(Z) + 2. In this case fc: Xc ---i C is a fibration by rational surfaces. The existence of a section is equivalent to t,he existence of a K(C)-point, on a surface X’ := X 8 K(C). The last, fact, is sufficient to prove for a nonsingular model X:,lin (over the field h’(C)). The surface X,‘rlin either is a de1 Pezxo surface or has a conic bundle struct,ure. In both cases Xirlir, has a K(C)-point. Note t,hat a gcneralizat,ion of Corollary 6.2.5 for the ca,se dim(X) dim(Z) > 2 is not known (Kolhir-Miyaoka-Tori (1992b)). Corollary Then

X

6.2.6.

is rationally

Let X be a nonsingular connected.

In the case dim(X) = 3, the Minimal criterion of rational connectedness. Theorem complete are

6.2.7 (Kollar-Miyaoka-hlori algebraic three-dimensional

Fan!o

Model

variety.

variety

Program

(1992b)). Then

with

p(X)

=

1.

gives the following

Let X be a nonsing,ula7 the following conxUions

equivalent

(i) H”(X; (Qi)a’“) = 0 for all n > 0, (ii) K(X) = -E and H”(X, QX) = 0 = H”(X, S’Q$) = 0, (iii) X %s rationally connected, (iv) X w b’zra t’zonnlly equivalent to eith,er a, Q-Fan0 three,fold with Q-factorial terminal singularities und p = 1, or a de1 Pezzo fibration over pl, or u conic bundle over a rational surface.

$6.2. Rational

Connectedness

of Fano Varieties

123

The implication (i) + (“)ii .is7 0 b mow . because both S”R‘$ and 0x(nKx) are direct surnmands in (ni)@m for some m. Frorn Proposition 6.2.3 we get (iii) + (i). Applying the Minimal Model Theory t,o X, we derive the implication (ii) ===+ (iv). To prove (iv) ====+ (iii), WC can use 6.2.5 and the following Proposition threefold nally

with

only

6.2.8 (Kollk-Miyaoka-Tori Q-factorG.zl singularities

(19921))). un,d

p(X)

=

Let X be a log-Fnno 1. Then X is ratio-

connected.

Proof. From 6.1.5 of singularities, and rationally corme(t~d, Then, by 6.2.5, 2 is

we get that X is uniruled. Let 7r: X’ + X be a. resolut,ion let f: U + 2 be the MRC-fibration for X’. If X’ is not then dim(Z) = 2 or 1. First, we assume that dim(Z) = 1. not a rational curve. Thus

hl(X,

Ux)

= hl(X’,

Up)

> hl(Z,

Uz)

> 0

(here we used the fact that log tcrmina.1 singularities are rat,ional, see 1.1.5). But on log Fano varieties one has Hr(X, 13~) = 0, a contradict,ion. Now. assume t,hat dim(Z) = 2. Since log terrninal singularities are yuotient,s of canonical singularities (Kawamata.-I%tsuda-Matsuki (1987)), then all tlrc> exceptional divisors EL of the morphism 7r are birationally ruled surfaces (Reid (1983a)). From Corollary 6.2.5 we get that, f: 1J ---f 2 contracts EL n U, whence for a srnall open subset 2, c 2 the morphism YT is an isomorphism onto f-r (&). Therefore there exists an effective divisor D c X which does not intersect f-‘(z) for z E 20. This is a contradiction with p(X) = 1. The

last proposition

Proposition dimensional ,nef

und

6.2.9 vuriety

hi,g.

admits

(KollbMiyaoka-Mori with

Then

a generalization.

X

only

is rutionully

log

(1992b)).

terminal

Let

singularities

Theorem be an

6.2.10

a non-empty

open

dimension point

(i) (ii) (iii)

(Koll&-Miyaoka-Tori Funo manifold,

n-dimensional and

z E Z

there

subset

C X,

a proper

surjecti,vr

exists

a rational

Theorems

of nonsingular rational

curves.

(1992c), Ca.mpana (1992)). char(k) > 0. Assume that there

u non~siny&r

projectke

morphism c?Lrue

6.2.10 and 6.2.4 we get

vuriety

r: U --j Z.

C on X

C meets V1(z), C is not contained %n n-l(z), -Kx . C < n + 1. From

he a three-Kx is

that

connected.

Below we discuss the problem of rational connectedness varieties of arbitrary dimension. To begin with we give the construct,ion of “horizontal” result is valid in any characteristic.

X

X

s?Lch

with

the

Then following

Z for

Fano This

Let eksts

of positive a generul psroperties

124

Chapter

6. Boundedness

and Rational

Connectedness

Corollary 6.2.11 (Kollar-Miyaoka-Mori (1992c), Campana. cha.r k = 0, then any Funo man$old is rationally connected.

(1992)).

If

We give a sketch of the proof of Theorem 6.2.10. As in 6.1, for a rational map 7r: X - - + 2, it is possible to define the notion of a relative deformation of a morphism f: C + X of a smooth project,ive curve C in X with a base set B c C such that f(B) c U. We denote the universal family of such deformations by Homz(C, X; f; B). Th ere , is an analogue of Proposition 6.1.1: if 7r is smooth along f(C), then the Zariski tangent space to Homz(C, X; f; B) at a point [f] is canonically isomorphic to H”(C, Inf*TxlZ), a,nd the obstructions to deformations are contained in H1(C, I,f *TX/Z). The following is a generalizat,ion of Proposition 6.1.9. Proposition 6.2.12 (Kollar-Miyaoka-Mori (1992c)). Let char k > 0, C be a smooth projective curve of genus 9 > 1, and let X be a nonsingular projective variety of dimension n 2 2. Let U c X be a dense open subset, and let rr: U + Z be a proper surjective m,orphism onto u nonsingular variety Z. Let B c C be a nonempty closed subscheme, and let f: C + X be a morphism such that f(B) c U. Further, assume that dirn[f) Homz(C, X; f; B) > 0, and let I be a smooth compactification of a curve I’ c Homz(C, X; f; B). Then the natural r&or& map F: C x I - - + X is n,ot a morphism. Furthermore, if

is a resolution of F, then every exceptional of the morphism rr o E: W + Z. Moreover, E on W with the following properties:

curve on W is contained in a fiber there exists an, ezceptional curve

E(E) is not a point and contains some point of f(B), E.F*H degf:(C), f’(C) contains some point Corollary exists

thesre

6.2.14. a morphism

(i) 7r 0 f = 7r 0 f’, (ii) dim[f,] Homi:(C,

of

if dim(Z)

> 0, then

if dim(Z)

> 0, then

of f(B).

Under the conditions of 6.2.12, f’: C + X such that X; f’, B) = 0

56.2. Rational We need some estimations relative) deformations.

Connectedness

125

of Fano Varieties

of the dimension

of the space of absolute

(not

Proposition 6.2.15 (KollBr-Miyaoka-Mori (1992c)). Let X be a Funo munifold, let T: U + Z be a proper surjectiue morphism defined on n dense open subset U c X, und let C be o, sm,ooth projective curIK of genus ,q > 1. Let B c C be a non-empty closed subscheme of degree b, and let ,f: C + X be a morphism slrch that f(B) c U. Th en there exists a constant 0 depending on,ly on (X, Z, T) such th,at dim[,l IIom(C, X; fi fs) > 0 wh,eneaer degrrof > a(b+g). The Step and a h: C + choose C’. By satisfies

proof

of Theorem

6.2.10 consists

of two steps.

I, char k = p > 0. Let us choose a. curve Co c X meeting x-‘(Z) point p. E Co n cl(Z). Let v: C’ ---i Co be the normalization, let C’ be the Frobenius k-morphism, and let, f = v o h,: C + X. We can h in such a way that deg7r o f > a(1 + g), where g is the genus of Corollary 6.2.14, we can replace f by the rnorphisrn f’: C + X which t,he following conditions:

1) cleg~of’=deg~of>a(l+g), 2) f’ has no relative deformations f (PO).

with

base scheme

B = {p},

where

p :=

But by Proposition 6.2.15, there are absolute deformations of f’. Then there are rational curves on X pa.ssing through ~0, which are images of exceptional divisors of the blow-up of C x r (see 6.2.12). Since deformations of f’ induce non-trivial deforrnations of 7r o f’, there is some exceptional divisor which is mapped onto a (rational) curve on Z. It is easy to check that this curve passes through z E Z. We fourld a %orizontal” curve on X meeting r-l (2). Further, as in 6.1.10, one can split, it into a sum of curves of degree < n + 1. Step II, lifting in characteristic zero. Since the anticanonical degree of “horizontal” curves is bounded, as in the proof of Theorem 6.1.4, then the lifting in characteristic zero is possible. For a different proof of Theorern 6.2.10, see (Carnpana (1992)). The following two assertions can be proved by using argument,s similar to those used in the proof of 6.2.10. Corollary 6.2.16 (KollBr-Miyaoka-nlori (1992c)). Assume that char k l;s urbitrary, and let ~1 X --i Z be a s,urject%,ue morphism between nonsingular projective varieties. Then (i) (ii)

nn,d dim(Z) > 0, then -Kx,z cannot be nm,ple, that in characteristic p > 0 th,ere is a resolution of sin,gulo,rities, and dim(Z) > 0, then -Kxlz cannot be ample except for the cuse when every fiber of T is sing&r. if T is smooth assume

Chapter

126

(5. Roundedncss

and Rational

Connectedness

Corollary 6.2.17 (Koll&r-Miyaoka-lori (1992c)). Assume that char Ic l;s arbitmry, an,d let nip:X 4 Z be a smooth surjective morphism between, nonsingular projective varieties. Assume th,at X is u Funo m,nn,ifold. Then Z is also a Fano manifold. From

6.2.10 and 6.2.3 (iii)

Corollary connected.

6.2.18.

WC have

Let X be a Fmo

manifold

over @. Thesn X is simply

Remark. The fact tl1a.t Fano manifolds are simply connected can be proved by another method. It follows from Calabi’s conjecture proved in (Yau (1978)) that a projective algebraic va,riet,y over @ is a Fano rnanifold if and only if on X there exists a Kahler metric with positive Ricci curvature. By (Myers (1941)) this gives us that the fundamental group of X is finite. So there exists a finite &ale cover 7r: X + X, where 2 is also a Fano manifold. But then with 2.1.2, see also x(0,-) = x(0x) . d e g: 7r > 1. This is a. contradiction of this fact see (Tsuji (1988)). (Kobayashi (1961)). F or a generalization In the rest of this (*)

section

we discuss

Let X be a (p ossibly singular) its degree (-Kx)‘” is bourlded

the following

problem:

Fano variety of dimension by a consta,nt depending

n. Is it true t,hat only on n.

In the two-dirnensional case WC know that, the degree of de1 Pezzo surfaces X (even with Du Val singularities) is bounded: Ii’; < 9. However for log de1 Pexzo surfaces this problern has negative answer. For example, let us consider over a rational normal curve (see 2.1.5 (vii)). Then X,, the cone X,1 c p”+l has only log terminal singularities, and Ki = 4/d + 4 + d + +w. But in fact the degree of log de1 Pezxo surfaces cannot grow too fast. It was proved in (Alexeev (1988), Nikulin (1989a), Nikulin (1988), Nikulin (1989), Alexccv (199413)) that t,he degrees of log de1 Pexzo surfaces are bounded by a csonstant which depends on indices or mult,iplicities of singularities. The first author (Iskovskikh (1979a)) proposed a few methods to bound the degree of Fano threefolds. The first of them actually uses the notion of rational connectedness. This method was later justified by Lvovskij (1981). For nonsingular Fano threefolds X, the inequality -KS < 64 wa.s obtained. This inequality obviously is attained for X = p3. Now this bomld also follows frorrr the classification of Fano t,hreefolds (see Chap. 4 and Chap. 7). Fano himself < 72. But, in his theory the variety X is (Fano (1936)) g ave the bound -K$ not necessarily nonsingular. There are some examples of singular Gorenstcin Fano threefolds with -KS = 72. However, if we assume that a Fano threefold X has only Gorenstein tcrrninal singularit,ies, then X is a deformatiorr of a smooth Fano tlrreefold (Namikawa (1997)), and in particular K: < 64. Example 6.2.19. Let F = Fo c p” be the inrage of the anticanonical embeddirrg of p”, and let W = Wg C p ‘(’ be a cone over FCJ. Then W is a

56.2.

Rational

Connectedness

of Fano

127

Varirtics

Q-factorial variety with a unique singularit’y at the vertex which is canonical Gorenstein. The anticanonical divisor on W is cut out by a quadric. The degree of W is equal to (-Kw):’ = 8 .9 = 72. Using Miyaoka-Bogomolov’s inequality for Chern classes of a stable vector bundle of surfaces of general type, Kawamata (1992b) proved the boundedness of degrees of Q-Fan0 threcfolds and the boundedness of indices of Q-Fan0 threefolds. The solution of problem (*) is known in the case of toric Fano manifolds. The bourldedness of the degree in this case was proved in (Batyrev (1982)). The case of log terrninal Fano varieties was studied in (BorisovBorisov (1992)): 1‘t was proved that if the discrepancies of X are > -l+~. then there are at, most a finite number of families of such Fano varieties of a fixed dimension. In particular, this gives the boundedness of toric Fano varieties with canonical singularities. The following simple lemma. shows that problem (*) is closely related to rational cormectedness. Lemma 6.2.20. For my ‘n, N E W there exists u constant c = c(n, N) such th,at if on a (possibly singular) n,-dimensional Fu,no variety X two general points x, y E X can, be connected b:y an irreducible curue of degree < N, then (j-&y


0, n: E X. For such rr,, k, a divisor D,r contains any irreducible curve C of D, contains any sufficiently general

6.2.10 one can obtain

Proposition 6.2.21 (KollAr-Miyaoka-Tori (1992c), Carnpana (1992)). Every two points on an n-dimensional Fano manifold X can be joined a chain of ru?ional curves of total degree < (2” - l)(n + 1). Using

Lemma

by

6.2.2, we get

Proposition 6.2.22 (KollBr-Miyaoka-Tori (1992c)). There is a constant N = N(n) such that u general point x on an n-dirnensionul Funo manifold X can be ,joined with every sufj%iently general pokt y E X by an irreducible cume of degree < N(n). Theorem For every n-dimensionul

r~

6.2.23 (KollBr-Miyaoka-Mori (1992c), Campana (1992)). E W there exists a constant c(n) such that the degree of an,y Fan,o manifold X does not exceed c(n).

128

Chapter

7. Fano

Varieties

with

p ) 2

Remarks. (i) The case p = 1 was studied in (Campana (199la), Nadel (1991)). Here, for c(n) there is an asymptotic formula c(n) - n”/4. However, it is expected that the sharp bound for (-K-g) is linear in 12. (ii) It follows from results of (Matsusaka that for every n, Fano manifolds of dimension nurrlber of quasi-projective schemes.

(1972)) and Theorem 6.2.23 n arc parametrized by a finite

(iii) The m&hod used in the proof of Theorem 6.2.23 can he applied to singular Fano varieties. For example (Borisov (1996)) proved t,hat all threedimensional Fano varieties with only Q-factorial log terminal singula.rities and p = 1 lie in a finite number of families whenever their indices arc bounded. (iv) The bolmd that can be obtained from the proof of Theorem 6.2.23 is not sharp. For example; it gives us the bourlds ( -Kx)~ < 9.2” and ( -Kx)” 5 8. 328 for n = 2 and 71 = 3, respectively. But the real bounds are (-h’~)~ < 9 and (-Kx)’ < 64, respectively. These bounds are achieved on p2 and I?“. However, for n > 4 the inequality (-Kx)” < (-Kp,,)“” = (n+ 1)” is not true. There are examples of toric Fano manifolds of the form X = pIpr,-~ (Opa7,-i “2 Up,, l(n)) with (-I(x)” > (-Kp!$)“; see (Batyrev (1981)).

Chapter 7 Fano Varieties with p 2 2 57.1. Fano Threefolds with Picard Number (Survey of Results of Mori and Mukai)

p 2 2

In this section WC shall consider only nonsingular Fano threefolds over an a,lgebraically closed field of chara,cteristic zero. For the classificat,ion of Fano threefolds with Picard number p = 1, see Chap. 4 (see also Chap. 3 and Chap. 6). Fano threefolds with p > 2 were Audied in (Tori-Mukai (1981), Xiori-Mukai (1983a), Mori-Mukai (1986)). In this section we shall give a survey of these results. Note that, these techniques can be generalized for the higher dimensional case (see Sect,. 7.2) and, proba.bly; for the case of positive characteristic by using (Koll6r (1994)). We note that Fano threefolds wit,h a structure of pl-bundle were classified in the work (Demin (1980)) , see also (Szurek-Wibniewski (199Oa,)). Toric Fano threefolds were classified in works (Batyrev (1981)) Watanabe-Watanabe (1982)) (all of them, except for IID‘, have Picard number p > 2). Lvovskij (1981) obtained the bound -K: 5 64 for the degree of Fano threefolds.

s7.1. Fano Threefolds

with Picard

Number

p> 2

129

Theorem 7.1.1 (Mori-Mukai (1981)). Let X be u Funo threefold with Picard num,ber p > 2. Then X belongs to one of the classes listed in Tables 12.3 12.7. Fano threefolds from the different classes cannot deform to each other. For each class from the tables, there exists a jIut family containkg all Funo threefolds from this family und which are pummetr-iced by an irreducible munifold. For classes No. 6, 8, 15, 23 in Tuble 12.3, the subclass (b) is a speciulizution of the subclass (o,). Corollary 7.1.2 (Tori-Mukai (1981), Mori-Mukai (1986)). (i) A Fano threefold X with -K$ = N exists if and only if N is an even nutuml number such that N 5 64 and N # 58,60. (ii) Let X be a Funo threefold with p > 2. Then the linear system 1 - Kxl contain,s u divisor which, is a sum, of two eflective divisors DI and D2. Furthermore, if 1- K x 1 as base point free (i. e. except for two threefolds No. 1 of Tuble 12.3 and No. 8 of Tuble 12.6), then D1 and 02 can be chosen in such a way that 1D1 1 un,d ID21 are also base point free. (iii) Let X + Y be a, blow-up of u Funo threefold of in,de.x r with p = 1 along u smooth (but possibly disconnected) curve C. Then X is a Funo threefold if and only if C is a scheme-theoretical intersection of divisors from the linear system 1 - ?Ky 1. (iv) The nsumber of classes of Funo threefolds up to deformations is given in the following table: P number (v)

of classes

2 36

3 31

4 12

By blowingup of points, it is possible threefolds from other ones: No. 14, Table I No. 28, Table

12.4 12.3

No. 19, Table 1 No. 30, Table 1 Q

5 3

6 1

7 1

to obtain

12.4 12.3

8 1

only

9 1

10 1

the following

No. 29, Table L No. 33, Table

211 0 Fano

12.4 12.3

The proof of the classification Theorem 7.1.1 is based on the study Mori cone and contractions of extremal rays on Fano threefolds.

VT ;3 of the

Definition (Mori-Mukai (1986)). A F a’no threefold X is said to be imprimitive if it is isomorphic to the blow-up of another Funo threefold with center ulong a smooth irreducible curve. Otherwise X is said to be primitive. Proposition 7.1.3 (Mori-Mukai (1986)). Let f: X + Y be a contraction of an extrernal ray of type El on a Fano threefold X, let D be the exceptional divisor off, and let C := f(D). Th en one of the following statements is true: (i)

Y is a Fano

th,reefold,

130

Cllapter

(ii)

7. Fano

Varieties

with

p 2 2

C 2 pl, und C has on Y normal sheaf of type (-1, -1) linear system 1 - mKy / is base point free for m > 0). This

Lemma tion 7.1.3, (i) (ii)

can l,e obta,ined

propositiorl

7.1.4 (Tori-Mukai we have

-KY C > 2g(C) - 2, if C r” pl: then -KY type (-1. -1).

from the following

(1986)).

IJn .d,PT the

C > 0, and equality

holds

pin this

simple condition,s

case

the

Lemma. o,f Proposi-

if and only

if C is o,f

Proposition 7.1.5 (Tori-Mukai (1986)). Let X be a Fano th,reefold Gth and let f: X + Y be a contraction of an extremal ray of type El. Then Y is a Funo th,reefold of index r > 2. p = 2,

Proof. Since Pit(Y) = Z, then Y is a Farm threefold. Assume that r = 1, and let C c Y 1)e a smooth curve which is the image of the exceptional divisor. Accordirrg to 4.2.2, on Y there exists a one-dimensional family of lines covering a divisor F (ample hecause Pit(Y) = Z). Therefore F C > 1, and if C is not a line, t,hen there exist,s a line ! on Y intersect,ing C. But, then for proper transforrn &’ := f -’ (!). we have -Kx Y’ < 0 which contradicts the ampleness of -Kx. Consider now the case when C is a line. According to 4.3.1, for g(Y) > 4 the linear system j ~ Kx 1 contracts some curves, and this is why it is not ample. In the remaining cases y(Y) = 2 and g(Y) = 3, it is easy to compute that -K: = 2y(Y) - 6 < 0, a contjradiction with the ampleness of ~ KY. Definition (Tori-Mukai (1986)). A n extremal ray R of type El is said to be of Qpe Eln if its conkruction f: X + Y gloves us a Fano th,reefold Y pi. e. we have case (i) of 7.1.3). Otherwise ((ii) of ‘7.1.3) R is said to be of type Elb. The following important t,heorern has a conic hundlr struct,ure. Theorem fold

with

7.1.6 p >

2.

shows that

(Tori-Mukai (1986)). Thesn on X there exists

each primitive

Let X un

Fano tjhreefold

be a primitive Funo threeray of type C (Cl or

extremal

C2). Proof. Assume the opposite. Then since p(X) > 2; on X there exist at least two extremal rays RI and R2 of types Elb, E2-ES, 01-03. Let f: X + y1, Z = 1, 2 l)e the corresponding contractions. If one of R,i, for example RI, is of type D, then p(X) = 2, and we have one of the following cases (according to 7.1.5, Rz canrrot be of type Elb): a) Rx is also of type D; then which is impossible,

t,he morphism

fl

x fz: X +

P’ x p’

is finite,

ji7.1.

Fano

Threefolds

with

Picard

Number

b) R2 is of type E2&E5, then fl maps the exceptional fz surject,ively onto lP1; this is also impossible.

p > 2

divisor

131

of the morphism

Therefore X has no extrema.1 rays of t#yptt D, and all extremal rays on X arc rays of type Elb or E2-E5. , D,, be the except,ional divisors, and let !!I, . : e,, be the correLet Dl,.. sponding extremal curves. of an ezLemma (Tori-Mukai (1986)). Let f: X + Y be a conhxction, tremal ray of type Elb or E2-E5, and let D be the exceptional divisor. Then -Din is ample, an,d every ejjfective divisor 071 D is movable.

It follows immediately from the lemma above that D; n D, = 8 (otherwise the curve S := D, n D,, is movable on D,, t,his is why S. D, > 0, but -D, 111, is ample on D,, so S.Di < 0). Further, WC take an effective curve C = (-mKx)‘: then C E C o,,B,: (I, > 0. Thus C. D, = ca,(iJ

D3) 5 0.

On the other hand, C. D, > 0 (by the arnpleness proves the t,heorern. We shall consider of conic bundle. Definition. variety

X

on,to

bundle if each or reducible).

now the case when

A morphism

f: X

(I n,onsingular

(71

fiber

off

of -Kx).

X is a Fano threefold

--f S

variety

to a conic

in

contradiction

wit,11 a structure

a nonsingular

from

~ I)-dimensional

is isomorphic

This

n-dimensional S is ca,lled

P2 (possibly

II

conic

non-reduced

Proposition 7.1.7 (see for example (Tori-Mukai (1986))). Let f: X S be a conic bundle over a projective varietr/ S, and let Z be a curve lying a fiber off. Th,en the following conditions are equivalent: a) f-l(D) is irreducible for b) p(X) = p(S) + 1, c) R=IW+[Z] 2,s an extremal Definition. sition

7.1.7,

Definition.

then

ray,

in

disvisor D c S,

und f is its contraction.

If the conic bundle f: X + S satisfies it is suid to be standard.

the conditions

of Propo-

The subset A,

is culled

each irreducible

+

the discrrmin,an,t

:= {s E S1 f-‘(s) locus of a conic

is not smooth} bundle

f: X + S.

Conic bundles over a higher-dimensional base will be considered in Sect. 7.2. Now, for the rest, of this section we consider only conic bundles .f: X + S over a t,wo-dimensional nonsingular and projective base S.

132

Chapter

7. Farm Varieties

with p > 2

Proposition 7.1.8 (see (Beauville (1977a), Mori-blukai (1986))). Let f: X + S be a conic bundle ouer a complete surface, let A, be its discrimino,n,t locus, and let Kxls = Kx - f * KS be the relative canonical divisor. Then, (i)

if AJ # 0, then A, is a curve with Sing(nf) coincides with the locus {s E S 1 .f-‘(s)

at worst

ordinary

do’uble points,

and

is not reduced}

(ii)

if A, = 8, and the surface S is rational, then f: X ---f S is a IID’-bun,dle locally trivial in th,e Zariski topology, (iii) ,4, E -f+K:,s, -4Ks G f,K: + AT, (iv) if X is a Fano threefold, then S is a rational surfacej (v) if c c A, as a smooth curve of genus zero, and Cn (A, -C> = 0, then the surface Xc := f-‘(C) M reduced an,d consists of two nonsingular ruled surfaces intersecting transversully along Some smooth section, (vi) if C C A, IC ic un irreducible component, and e is the closure in X of the locus of singular points off -l(s), s E Greg, then K x,~. where

n is the number

C = n + :(A,

o,f ordkary

Assertions (i), (“‘) m are well-known, Br(S) of a rational surface is trivial.

double

- C) points

C , on C

(ii) uses the fact that t,he Brauer For (iv), see Proposition 7.1.11.

group

Lemma 7.1.9 (Sarkisov (1980)). Let f:X + S be u conic bundle over a nonsingular projective surface S, and let C c S be an irreducible curve such that the surface XC := f-‘(C) is reduced. Then (i) C is a nonsingular connected component of Af , (ii) Xc = El U E2, where E; + C = IP1 is a @bundle for i = 1, 2, (iii) there exists the following commutative diagram below

(iv)

2

133

Under tl1c conditions of Lemn1a 7.1.9, we say that tl1e conic burldles f,: X, + S, i = 1, 2, are obtained from each ot,her by an elementary tmnsformation wit,h center along the curve C,. For a co11ic lxmdle f: X + S, it is easy to compute the topological Euler characteristic and tl1e Bet,ti numbers. Lemma 7.1.10. Let f: X + jective surfuce. Then

bundle

xto,,(X) = 2(Xtop(S) - Pa(Af) + 11, hi(X) = 131(S), b3(X) = 2(h1(S) + b(X)

(i) (ii) wher,e

pU(Af)

is th,e

arithmetic

genus

Definition. A conic bundle Funo conic bmdle (respectively, Proposition

7.1.11

phisrn that

with

FrL the

notation,

over a nonsingular

- b(X)

of Af

pwe

+pa(Af)

assume

pro-

~ 2),

that

pn(@)

= 1)

(respectke[y, p’ -bundle) f: X + S is culled Fano pL-bundle ) if X is a Fano variety.

a

(Mori-Mukai (1986)). (i) Let f:X + S be a rnor‘fibers onto a nonsingular surface S. Assume threefold. Then f: X + S is e%th,er a Fano conic bundle

one-dimensional

X is a Fano or Fmo @-bundle.

(ii)

S be a conk

as

above, if X is Q Fano

threefold,

then

S is u

de1 Pezzo

surfuce. Outline of proof. (i) Since -Kx is ample, tl1en -Kxls is relatively amand have ple. Since fibers of tl1e morphism f: X + S are one-dimensional ample anticanonical divisor, they are nothing but tonics in p”. (ii) By the numerical Nakai criterion of ampleness, it is sufficient to prove that some multiple of -KS is rlumerically equivalent to an effective divisor, a,nd tl1at -KS C > 0 for any irreducible curve C. By Proposition 7.1.8, (iii), we have -4Ks I f,(K$) + A,. For sufficiently large rn, tl1e divisor -rnKx is very ample. Tl1en -4na’Ks G f * (HI Hz) + m2A,, where HI , Hz E I- rnKx 1, and HI Hz is an irreducible curve. If C $ A,f , then

-4Ks. If C 11ave

c

A,

-KS.

is an irreducible

C = f*(-Ks)

Tl1e proposition Proposition Lerrmu 7.1.9, (i) (ii)

C = f,(-Kx)2.

C + A,. component,

‘6

= -Kx

C > (-Kx)2. then,

f*C

by Proposit,ion

. c = -Kxls

‘6” > -Kx.

> 0. 7.1.8,

(v), WC

c > 0.

is proved. 7.1.12 (Mori-Mukai (1986)). Un,der ussurne thut X %s a Fano 0 because the curve B is movable on E2 ‘v This contradiction proves (ii). Proposition 7.1.13 (Mori-Mukai bundle! and let E be a (-1)curve is imeducible. Then (i) fix,: (ii) there

XF: + 5’ is n PI-bundle, exists

uhere

(Y: S

f’:

+

X’

section,s

S’

the

commutative

+

S’

is

is a conic

of the

morphism

(1986)). on 5’ such moreover diagram

the

conkactionT

Furt,her, El n E2. El < 0. P1 x i?‘l.

Let f: X + S he a Fmo conic that the surface XE := f-‘(E)

XF: = p1 x @, below

of

the

exceptional

curve

E’

on

S,

bundle, CT: X’ + X is a contmction of horizontal f lx*: : XE ---) S, while f = f’ x&71 S.

Proof. Let 2 c X be a curve such that f (2) = E. Since NE(X) is a rational polyhedron generated by a finite number of extremal ra,tional curves, say Li, . :e ,,,, we have Z = c a,!~, a,; t Q, a,; > 0. Since 7r,Z . E < 0, then ?r& E < 0 for some i. This gives us that there exists an extremal rational curve Il = .Ci which generates an ext,rernal ray R and such that, *f(e) = E. Since e. f-‘(E) < 0: then R belongs to one of the t,ypes El, . , E5, and divisor for R. Since XE has a surjective XE = r-l(E) is th e exceptional morphism onto E ‘v p’, then R cannot, be of t,ype E2, E4 or E5. Moreover, the fibers of XE + E are not numerically equivalent to e, therefore R cannot be of type E3. So we have only the case when R is of type El. Therefore XE has another structure of p’-bundle. Whence XE: Y P’ x @. This proves (i). The contraction a = CR: X + X’ of the extremal ray R again gives us a Fano threefold because the normal bundle NxEix ‘Y f *NEis is not negat,ivc (see 7.1.3). It is now clear that there exists a morphism f’: X’ + S’ which is a conic bundle such that f’ ~7 = o f: and f = f’ x S, S. The proposition is proved. Corollary 7.1.14. (-l)-curve, then component of A,f

SW a

One of the main is the following.

Let f: X 4 S be a Funo conic bundle. Th,en if E c S either E does not irkrsect A, o*r E is a connected

points

in the classification

of Fano t,hreefolds

with

Q> 2

Theorem 7.1.15 (Mori-Mukai (1986)). Let f:X + S be n. Fano bundle, and S $ p2, S $ IF1 and S $ p1 x p’. Then f is trivial, X E S x B1, md f is the projection onto the first factor.

conic i. e.

s7.1. Fano Threefolds The first step in the proof

wit,h Picartl

of this theorem

Number

p > 2

135

is t,hc following.

Lemma 7.1.16 (Mori-Mukai (19%)). Let f: X + S br a Farro co,n%c bundle, und let S $ IF’“, S y? IFI usnd S $i IP1 x P1. Then f *is n p1 -bundle. Proof. Since S is a de1 Pesxo surface (see Proposit,ion 7.1.11, (ii)) and S is not p2, IF1 or p’ X p’ , then there exists a hirational morphism (1: S 4 p’ X P’ which is the blow-up at points .x1, : :I;,, E p1 x pl; where n = p(S) ~ 2 > 1. 2, = (y7.zl), and let L,; A& be the proper transforms of Set E, := fl-l(:r,), lines ‘J, x @ and p1 x z, on S, i, = 1, . . , ‘n. Then E,, M,, L, are (-1)-curves on S. We claim that E,; At,, L, art not contained in A, and do not intersect n,f for all i = 1, : 7~. Indeed, assume that A, n E, # 0. Then by 7.1.14 the curve E, is a connected component of ni. Therefore A, n L, # 8, and L, also is a connected component of A,, which is impossible. This gives us AJ f’ E, = 0 and sirnilarly A, n L, = 0, Al n M, = 0. Therefore n,f is contained in S \ (UE,) \ (UL,) \ (Unfi). Because this loc*us is an affine surface, and A, is compact, WC get A, = 0. The lemma is proved. So the proof

of Theorem

now is reduced

Lemma 7.1.17 (Mori-hlukai let u,: V 4 T be a ~l-bundle.

(ii)

(1986)). Then

to the case when Let p:T

-

p’

f is a p’-bundle. be a pl-bmdle,

and

if V is n Farm threefold, then there me only two possibilities: q~, a) ‘$)-l(F) = @ x IP1 for each fiber F of the morphism p. 1,) lj-‘(F) e IF 1 f or each fiber F of the morphism

Proof. (i) Because the surface T is rational, its Brauer-Grothcndicck group Br(T) is trivia,l. Therefore V is isomorphic t,o IT f or some vector bundle of rank 2 on T. Consider the natura.1 morphism 7: p*‘p*l + E. By the base change theorem and by our assumption, p*E is a vector bundle of rank 2 on i?’ and 7 is an isomorphism on every fiber F of the morphism p. Therefore 7 is an isornorphism, and p(&) = I xp T. (ii) A l%bundle $-l(F) is a fiber of the map q~ o $: V - PI, a.nd by t,htl adjunction formula it must be a de1 Pexxo surface. Since p’ xl?’ and IFI cannot be deforrned to each other, we have that there are only the two possibilities a) and 1)). ProoJ (of T/~,eorern ?‘.1.15]. As has been shown above, we may assume that f is a ~l-l~uridlc. Let (1, 7~~) zi, Lf, M; be as in the proof of Lcmrna 7.1.16. Applying Proposition 7.1.13 step by step, it is possible to show t,hat there exists a Fano l?-bundle f’: X’ + p1 x p1 such that, f = f xpxp a. Since, by Proposition 7.1.13, fldml(l,l) and f lSpl(nr,) are trivial p’-bundles, so are f’I,f,--l(zY,x~l~ and f’lf,-lpXZ1). Therefore by (ii) of Lemma 7.1.17, the restriction f’ls,+l (,Vxpl 1 is a trivial lKburrtlle for each point. So there

136

Chapter

7. Fano Variet,ies with p > 2

exists a @-bundle fo: X0 + P1 such that f’ = fc x (lP1 x P’). Since Xc = fel(P1 x ~1) r” Pr x IP1, then fo is trivial. This gives us that the IP-bundles f and f’ are also trivial. Theorern 7.1.15 is proved. Theorerns 7.1.6 and 7.1.15 describe completely the stjructure of primitive Farm threefolds with p > 4. To classify imprimitive Fano t,hreefolds, it, is sufficient to describe curves on Fano threefolds X such that blow-ups X’ + X of them give us a new Fano threefold X’. Consider this question in detail. Lemma 7.1.18. Let f: X’ + X be the blow-up of a smooth curue C on a n,onsingular projective three,fold X, and let Z he a smooth irreducible reduced curve on X. Then (i) (ii)

if -Kx if -Kx

f Z = 1, then either Z = C or Z n C = 0, Z = 2, then eith,er Z = C, Z n C = 0 or C

n Z con&& of one point in which the curves Z und C intersect transversally.

Proof. Assume that -Kx 2 < 2, Z # C and Z n C # 0. Let Z’ be the proper transform of Z on X, and let, E = f-l (C) be t,he exceptional divisor of the blow-up f. We have 0 < -Kx/

Z’ zz f*(-Kx)

Z’ - E. Z’ z -Kx

. Z - E. Z’.

Since Z # C and Z n C # 8, then Z. E > 0. Whence -Kx . Z > 2. This us that, -Kx . Z = 2 and E 2 = 1. This proves the lemma.

gives

Lemma 7.1.19 (Tori-Mukai (1986)). Let f: X’ + X he th,e blow-up of a smooth curve C on, X. Assume that C Y P1, an,d -Kx C = 1. If X’ is a Fan,o threefold, then NC/X r” 0~1 gi O~I (-1).

Proof. Since deg NC/X = -Kx C + 2y(C) - 2 = -1 (see Lemma 2.2.14), then NC/x r” 0,1(n) @ &1(-l - n). L e.t s be the negative section of the exceptional divisor E p p(N ~1~) of the blow-up f. Then -Kx .s = 1 -n > 0. We get n = 0. To classify imprirnitive Fano threefolds with p > 3, it is sufficient to describe curves on Fano conic bundles such that blow-ups of such curves again give us Fano threefolds. Proposition 7.1.20 (Mori-Mukai (1986)). Let g: Y + S he a conic hundle, and let C he a smooth irreducible curue on Y. Assume that th,e blow-up 0: X + Y along C is uyain a Fun,0 threefold. Then (i) (ii)

C does not intersect singular fibers of g, one has on&y the following possibilities for C: a) C is a smooth fiber of g, and X is a conic bundle ouer S’, the blow-up of S at the point g(C), b) C is a regular su.hsectionof g ii. e. g/c: C + S is an embedding); in this case f = .(I . CT:X + S is a conic bundle with discrkinant locus A, = Ag u g(C).

s7.1.

Fano

Threefolds

with

Picard

Number

p > 2

137

Corollary 7.1.21 (Mori-Mukai (1986)). Let g:Y + S be a Fano conic bundle, usnd let h: X + Y be a composition of blow-ups along smooth cwws such that X is a Fano threefold. Then there exists the following comm,uta,tive diagram h x

where ups.

f: X + S’ is a conic

Corollary 7.1.22 bundle, an,d let C be locus A, is an ample a Fano thsreefold, then As an immediate have

-

bundle,

X’

and Q: S’ +

S is u composition

of blow-

(Mori-Mukai (1986)). Let g: Y + S be a Fano conic a smooth irreducible curve on Y. If th,e discrim,inant divisor on S, and the blow-up X + Y of Y along C is C is a smooth fiber of g.

consequence

of Theorem

7.1.15 and the last corollary,

we

Proposition 7.1.23 (Mori-Mukai (1986)). Let g: Y 4 S be a non-trivial conic bundle over S = F1 or P1 x IP”. If every curve on S intersects A,, then there are no blow-ups of Y along a smooth curve which ure Fano threefolds. Proposition 7.1.24 (Mori-Mukai (1986)). Let g: Y 4 S be a Fun.0 conic bundle, und let C be an irreducible regular subsection of g. If th,e blow-up oj Y along C is a Fano threefold, then an elem,entar?/ transformation g’: Y’ + S satis&es one of the following conditions: (i) Y’ also is u Fano (4

C - pl,

dn-~(~l(c))

threefold, is a trivial

P1-bundle,

and -KY/s

C = 2g(C)

+ 1

Further we show how the theory developed above can be applied to the classification of Fano threefolds with p = 2. Let X be a Fano tjhreefold with p(X) = 2. In this case the Mori cone NE(X) c N1(X) E JR2 has exactly two extremal rays. Denote them by RI and Rz. Let f,: X ---f Yi be the contraction of R,, let & be the corresponding extremal rational curve, and let pi = p(R,) be t,he length of the extremal ray R,t, i = 1,2. Let, L; be an arnple generator of Pic(k;), and let Hz := f,,*L,, i = 1,2. The following theorem plays the central role in the classification of Fano threefolds with p = 2. Theorem (9

7.1.25

(Mori-Mukai Pit(X)

{HI, Hz} Moreover,

(1983a)). = f; Pic(Y,)

is u basis of Pit(X), - Kx - ~2H1 + ~lH2.

Notation @

and {,!I, !,}

f;

as uboue.

Then

Pic(Yz), is the duul

basis of N1 (X).

Chapter

138

(ii)

If R2 is of

Let X is a blow-up So we By

E2;

type

f‘j -ezception,al

7. Fano Varieties

E3,

E4

or E5,

with p 2 2 02

then

PI = 1. ushers: 02

IIS prove for example (i). If one of the rays RI, R2 is of type 03: then p2-bundle over pr, and then X is isomorphic eit,her to p1 x p” or to the of p’ along a line. In these cases assertion (i) can be checked directly. shall assume that RI, R2 arc not of type 03. (1.3.1) and 1.4.4, there a.re exact sequences 0 i

Pic(Y,)

J+

Pit(X)

(2

Z -

0,

i = 1,2.

By the definition of extremal contractions, we have f; Pic(Yr) (0). Let CLbe the order of the quotient group Pic(X)/f;

Pic(Yr)

U-R-x It is sufficient

) = p2H1

to show t,hat CL= 1 This 24~1. = -aKx

By the Riemam~Roch

Lemma 7.1.26

c2(X) theorem

=

(7.1.1) n f; Pic(YJ)

=

8 f2* Pic(Y2).

It follows from the exact sequence (7.1.1) that 1-l, l; = 0 and -Kx & = /L, for i = 1, 2. then

X.

%s the

dilJiSO?-.

Hr

&2 = Hz . F1 = a. Since

+ p1H2.

can be obtained

P2Hl

c2(X)

+

p1H2

from

the equality

c2(X)

we have the following

(Tori-Mukai

(1983a)).

D = cix(Ou)

+ cix(UD(D))

Let D

be an e,fle:etioe divisor

on

Then c2(X).

Using the above lemma can obtain the following

and the classification table

- 20”

- (IQv$

of extremal

. D.

rays (see 1.4.3),

orre

ji7.1.

Fano

Thrrefolds

with

Pimrd

Number

p > 2

139

in which T’ is the largest integer dividing - KX +D, (respectively, -Kx + 20; 3 2(-Kx) + Di) and D; is the exceptional divisor of Ri in t,he cases when R, is of type El. E3 or E4 (respectively, E2 or E5); C is the center of the blow-up fl, and dcgC = L, C if R, is of type El; A is the discriminant locus of t,he conic bundle fi if R, is of type Cl; and X, is a general fiber of fl if R, is of t,ype Dl. For example in the case when R, is of type E3 or E4, by Lemma 7.1.26. + Di z r/H,, then Q(X) H,, = 24/r’. we get cz(X) D, = 0. 5‘incc -Kx Further assume that K, is of type Cl. Then Y, p p”, and. by the formula in Proposition 7.1.8, (iii), we have deg A < 11. Whence 7 < c2(X) Similarly, Theorem

. H,, < 17.

(7.1.2)

if R, is of t,ype Dl , then by the classification 1.4.3: wt: ha,ve 1 5 (Kx,)’ < 6, and therefore 6 5 c2(X).

of ext,remal

rays (see

H, < 11.

(7.13)

Let us prove (i) of Theorem 7.1.25 in the case when X is primitive (the imprimitivc case can be treated in a, similar way, see (Mori-Mukai (1983s)). By Theorem 7.1.6, there exists an extrrrnal ray, say RI. of type Cl or C2. Then the second extremal ray RZ is of type E2, E3, E4, E5, Cl, C2: Dl or D2. From the table and (7.1.2)) (7.1.3), WC’ get Q(X). One has t,he following

Hz < 24

or

= 45.

(7.1.4)

possibilities:

1) RI is of type Cl, a) 1~2 = 1, a(-Kx) G HI + Hz. If Q(X) . H2 = 45, therl, by (7.1.2), (7.1.3), we have 51 5 24~ < 62, a contradiction. From (7.1.4) we get Q(X) H2 5 24 and 240, = Q(X) . HI + Q(X) H2 < 41, i. c. u = 1. b) /LZ = 2, a( -Kx) z 2HI + Hz. Since H2 is a primitive element, of Pit(X), then CLis odd. It, follows frorn t,he classification of extrcmal rays and ~2 = 2 that, Rz is of type E2: C2 or 02, and from the t,able above we get, Q(X) 112 < 24. Therefore 240, < 58, i. e. a, = 1. 2) R2 is of type C2. This case is t,rrated in a similar way, see (Mori-Mukai (1983a)). Thus part (i) of Theorem 7.1.25 is proved. Now, we show how Theorem 7.1.25 ca,n be applied to the classification of Fano threefolds with p = 2. Let us consider for exa.mple the ca,se when both the extremal rays RI: R2 are of types Cl or C2. Then fl, ,f2 are conic bundles over p’. By the definition of extremal rays, the morphism ,f = (fl, fz): X + @ x p” is finite. Lemma (Z/P27

VPl).

7.1.27

(Alori-Mukai

(1983a)).

.f*X

is

II, &,&or

oj’

bidqree

Chapter

140

Proof. Put h& := $&~(l), jection onto the i-th factor. Theorem 7.1.25 we have f+X. 2/m

Thus

= (2/~~)&1i

Lemma birational,

f*w-f(x)) formula

7.1.28 then

nf;

+ (2/pi)nlz.

with

p ) 2

(Mori-Mukai

. 442 = f*Aff

The lemma (1983a)).

If

. f*nJz

= Hf

Hz =

is proved.

the

morphism

f:X

+

f(X)

is

it is an isomorphism.

Assume

that f is birational. Since f is also finite, then -h’x N where n is an effective divisor on X. By the adjunction + 4 and by Lemma 7.1.27,

f*(-Kf(x)) Therefore, by Theorem The lemma is proved. Now:

Varieties

where YT~:P’ x p2 ---f p2, i = 1,2 is the proSince f*fVi = Hi and H,f E (2/l~i)&, then by

((point) x (line)) = f*X (e, . Hz) = 2/w.

f*X

Proof.

7. Fano

for the pair

Case (Cl,Cl). f(X) is birat,ional, divisor of bidegree not birational, then bidegree (1,l): and Case (Cl,C2). (No. 4, Table 12.3).

7.1.25,

(RI, Rz),

= PZHI + plffz.

n = 0, i. e. f: X -+ f(X)

one has the following

is an isomorphism.

cases.

f*(X) is a divisor of bidegree (2,2) (see 7.1.27). If f: X then according to Lemma 7.1.28, X is isomorphic to (2,2) in p2 x YY2 (No. 6, Table 12.3). If f:X 4 f(X) this is a double cover, f(X) is a nonsingular divisor we again get a Fano threefold No. 6, Table 12.3. X is. isomorphic

to a divisor

of bidegree

(1,2)

4 a is of

in p2 x p2

Case (C2,C2). X is isomorphic to a divisor of bidegree (1,l) in p2 x IID” (No. 32, Table 12.3). Other possibilities for ext)remal rays RI, R2 can be considered in a similar way. The classification of primitive Fano threefolds with p 2 3 can be obtained from the following Theorem fold. Then 6) (ii)

(iii)

P(X) ifp(X)

2fp(X) divisor structure

7.1.29

I

(Alori-Mukai

3, = 2, then X = 3, th en X

(1983a)).

is a conic

Let X be a primitiiue

over p2, bundle over IF1 x P’ D E IID’ x P’ s,uch that 00 (D) N c3p xp (-1, -l), of conic bundle over p1 x @. either

Fano

three-

bundle

is a conic

und contains a or has another

57.2. Higher-dimensional

Fano Varieties

with p > 2

141

Indeed, by Theorem 7.1.6, on X there exists an extrema.1 ray R of type Cl or C2, and its contraction cp: X + 5’ is a conic bundle such that p(X) = p(S) + 1. Further, by 7.1.13 and by 7.1.8, (iv), S 1s a minimal rational surface, and by 7.1.11 (ii), S is a de1 Pezzo surface. Therefore S is either p” or p’ x @, and p(X) = 2 or 3. To prove (iii), assume that, p(X) = 3. Let us consider an extremal ray R’ # R. Suppose that RI is of type E2, E3, E4 or E5, and let D be the corresponding exceptional divisor. In cases E2 and E5 we have D 2 P2, and D ha,s no non-trivia.1 rnorphisms onto p’ x pl. Therefore ‘p(D) is a point on S, a contradiction with the definition of extrernal rays. The case of E4 is considered in a similar way. If R’ is of type E3, then D 11 p1 x @, and the generators are numerically equivalent on X. This gives us also that v(D) is a point. Thus X does not contain extrernal rays of types E2, E3, E4 and E5. Since X is primitive, then R’ can be of type Cl, C2 or Elb. The theorem is proved. We give the following statements without proofs. For deta,iled proofs, we refer the reader to (Mori-Mukai (1983a), MoriMukai (1986)).

fold

Proposition with p(X)

(Mori-Mukai (1983a)). Let X he un imprimitive = 3. Then one of the following statements holds:

(i) X is a blow-up (ii) X is a blow-up Of tU10

(ii)

c

p4 along

three-

a disjoint

union

CILI-lJl/f;'S.

Theorem 4. Then (i)

of a conic bundle ‘p: Y + p”, of P” or a smooth quad+ Q

Funo

(Mori-Mukai

(1983a)).

Let X be a Funo

threefold

with

p(X)

>

p(X) = 4, then X h as a conic bundle structure cp: X + S such that one of the following holds: a) S$pl x@, b) S y p1 x p’, and the discriminant curwe A, c S is umple. If p(X) > 5, th,en X has a conic bundle structure ‘p: X + S such thd If

SyiAlP

XP.

In the conclusion of this easy to obt,a.in t,he following Theorem (Mori-Mukai 6. Then X N S11-, x pl,

section

we note

that

from

Theorem

7.1.15

it is

(1983a)). Let X be a Funo threefold with p(X) > where S11-,) is a del Pezzo surface of degree 11 -p.

57.2. A Survey of Results about Fano Varieties with Picard

Higher-dimensional Number p > 2

In the present section we shall consider the problem of classification of nonsingular Fano manifolds X such that dim X > 4 and p(X) 2 2. As in the

142

Chapt,er

7. Farlo Varieties

wit11 p > 2

case dim X = 3 and p(X) 2 2, the known result,s obtained in this direction are based on the technique of contraction of extrcmal rays. The most complete results are obtained for Fano manifolds of la,rge (with respect to the dimension) index. Mukai in (BirationalGeometry (1988)) posed the following. Cmjecture

Picard (Fly.

number

7.2.1. Let X be a Tj,-dimensional Fano manifold of index T with p. Then T < n>/p + 1, and the equa.lity holds only for X E

From the classification of Fano threefolds it is c1ea.r that Conjecture 7.2.1 is true for n = 3. In the works (Wisniewski (1990b), Wibniewski (1991a)) Wibniewski obtained results which imply Conjecture 7.2.1 for p = 1: p = 2 and any n. We discuss these results below (Theorems 7.2.1 and 7.2.2). Theorem manifold

7.2.1

(W’ lsniewski

(1990b)).

Let X

be an, n,-dimensional

Funo

0.f inxiez r>

Then p(X)

$+I.

= 1.

Proof. Let H be a fundamema.1 divisor on X, let Co be a rational curve on Denote by X such that H Cc, = 1, and let f: P1 + Ce be its normalization. dimension (see 6.1). U a component of the schernc Hom(P1, X; p) o f maximal By the Riemann ~Roch Theorem and 6.1.1, we have dim U > x(pi,

f*Tx)

= 71 + r’.

Further, let Chowh(X) be the Chow variet,y pa.rametrizing effective 2 on X such that 2 H = 1 (not,e that any such cycle is irreducible H is ample). There exist,s a natural morphism

l-cycles because

U ----f Chow;,(X). Let T be its image. Then T is an irreducible (1979)) Wisniewski (1989a)): and dirnT There

= dim U - dim Aut(Pr)

is a diagram V-T

PI X

Y

projective

variety.

> ‘II + r - 3.

see (Mori

(7.2.1)

s7.2.

where

Higher-dimensional

Fano

V c X x T is the incidence

Varieties

p > 2

143

variety:

V := ((2, t) 1 IC lies on a curve, and p, q a,re projections.

with

Since all fibers

paramet,rized

by t},

of q are of dimension

1, WC have

dim V = dim T + 1 2 71 + T - 2.

(7.2.2)

It also follows from the construction that p(V) is a closed subset in X. Rational curves C on X with the propert,y -Kx . C 5 n + 1 cover a dense subvariety in X, see (Nori (1979), Miyaoka-Mori (1986)), and also 6.1.10). In our case, since 2r > n + 2, this means that rational curves on X with the property C . H = 1 cover a dense subvariety in X. Thus, we can choose Co and U in such a way that p(V) = X. Further, suppose that p > 2. Then there exists an extrcmal ray R on X with the property [Co] @ R. The length p(R) of the extremal ray R is divisible by r, this is why each non-trivial fiber of the contraction cp = contrR: X + Y is of dimension > T - 1 (see 1.4.6). Let F be an irreducible component of such a fiber of maximal dimension. Since p(V) = X, there exists a rational curve C from the family parametrized by T such tha.t C n F # 0. But, [Co] = [C] @ R, so C @ F. Let us choose a point x E C, x 6 F, and let T,. be an irreducible component of q(p-’ (z)) containing the class of C. Obviously, we have n F # 0

P(q-yrIJ) The restriction

is finite Further

(7.2.3)

morphism

everywhere

outside

p-l(~),

see (W iGniewski

dim(q(p-‘(TT))

(1990b),

Ionescu

n F) < 0

( 1986)). (7.2.4)

Indeed, ot,herwise this intersection contains some curve B. Ta.ke the normalization and the base change. We get a ruled surface 7r: S + B and a morphism $: S ----f X (the composit,ion of p with the base change and with the normalization). The map li/ cont,ract,s some section So to a point 2, and Q(S) intersects F along a one-dimensional set. Put F. := y’lV1 (S n F). We get, t,wo disjoint curves Bag and Fo on a ruled surface S with negative index of self-intersect,ion, which is impossible. This proves the inequality (7.2.4) (a similar technique was applied by Miyaoka-Mori (1986), see also Sect. 6.1). We shall complete the proof of our theorem. From (7.2.3) and (7.2.4), we have dimp(q-‘(T,)) + dim F < dimX, i. e. dimp(q-‘(T,)) But,

on the other

hand,

dimp(q-‘(T,)) a contradiction.

using

(7.2.1)

= dimq-‘(T,.)

< n - T + 1. and (7.2.2), = dimT,

we obt,ain + 1 2 r’-

1,

Chapter

144

Remark. pseudo-index

Theorem

7. !&no Varieties

7.2.1 will

remain

T’ := mire{-Kx Theorem n-dimensional Then,: (i) (ii)

7.2.2

C

manifold

of

Zf n = 2r ~ 2, then X E IF-’ 2

(1991a)). Let X be an (1990b), W’ r4niewski index r > 2 with p(X) > 2 and n < 2sr - 1.

(Wibniewski

Fano

with

(0~2

of P2rP1

of the of

bidegrxe ( l))@(‘pl)

(2) @c3,2 along

tangent

quadric, bundle or, what is (1, l), or, what is the same,

I?‘-“.

The sirnplest and the most studied class of Fano manifolds with the class of Fano manifolds with a projective space burrdle structure. Definition. A morphism called a projective space equivalent conditions h,olds:

bundle

(i)

is isomorphic

is

(ii)

each

fiber

sion,

dim X -dim

yP1

(y),

y E Y I”,

is the

From

projection

Theorem

Corollary

Y of nonsin,gular (or

scroll)

and the morphism,

open subset U C Y, X is the projectiuization Y

‘p: X +

IPy (6) onto

the

if

projective on,e

to the projective p has

of some vector

a section,

bundle

of

the

space over

/I > 2 is

manifolds

following

two

IF’ of dirnensome

Zariski

& on Y, and cp: X --f

base.

7.2.2 WC get

7.2.3.

Under

the condit%ons

of

Theorem

7.2.2,

the variety

X

is a scroll.

Fano manifolds with a st,ructure of projective space bundle are studied by the vector burrdle method. Note that it, follows from 6.2.17, see a.lso (Wibniewski (1991b)); that if a, nonsingular Fano variety can be represented in the form P+(E), t,hen t,he base variety is also a nonsingular Fano variety with p(Y) = p(X) - 1, and the projection ‘p: X + Y is a contraction of a.n cxt,remal ray. At, the present time complete results in this direction are known only in the case Y = P2: Szurek-WiSniewski (1990d) obtained the complet’e classification of Fano rnanifolds of type PPa (E) where E is a vector bundle of rank T > 2 on L?‘. In this case the classification contains 8 types of such manifolds. There are also some particular results for four-dirntnsional Fano manifolds of type Py (E).

57.2.

Higher-dimensional

Fano

Varieties

with

145

p > 2

Theorem

7.2.4 (Sxurek-Wibniewski (1990b)). Let X be a Fano fourfold X can be represented in the form pp’(E), where & is a vector bundle of rank 2 on p’. Asssume that cl (&) = 0 or -1 (we can always get this situation by twisting & with a line bundle). Then & is one of the following:

such

that

6) &=Uc9U, E = u 830(-l), E = U(-1) CBU(l),

(ii) (iii)

8 U(l), (iv) & = U(-2) & %s the null-correlation vector bundle on p” (the kernel of the (VI phism Tpd (- 1) + 01p:s(1)) see (Szurek- Wiinniewski (1990b)).

epimor-

Theorem

7.2.5 (Szurek-Wibniewski (199Oc)). Let X be a Fano fourfold X can be represented in the form, po(&), where & is a vector bundle of rank 2 on a smooth three-dimension,al quad% Q c IP4. Assume that cl(&) = 0 or -1. Then & is on,e of the following: such

(i) (ii) (iii) (iv)

that

& I E &

= = = is

0 CT3U, 0(-l) CEO(l), 0 CB 0(-l), a stable bundle with q(E) = 0, cz(E) = 2; each s,uch bundle is the pull-ba,ck of the bundle from, (v) of Theorem 7.24 under a finite cover of degree two Q + p’, (v) & is a spinor bun,dle on Q, i. e. the restriction of the universal bundle of rank 2 over the Grassrnannian Gr(2,4) to its hyperplane section Q c Gr(2;4) c P’. Here q(E) = -1, CZ(&) = 1.

Theorem 7.2.6 (Wihiewski (1989b)). Let X be a Fano fourfold of index 2 such that X can be represented 1;n the form &(&), where E is a vector bundle of ma,& 2 on a nonsingular th,reefold Y. Th,en Y is one of the following: (i) X = Y x p’, where Y is a de1 Pezzo threefold or Y = p”, (ii) X = Pp(Up(l) @Up-l) o*r X = ~Q(UQ @ UQ(-l), where Q c p4 is a smooth three-dimensional yuadric, (iii) X has two structures of PI-bundles, and it is the variety from (v) of Theorem 7.24 or from (v) of Th,eorern 7.2.5. We now state some facts about cont,ractions of dimension > 4 (see also 1.4.5- 1.4.8).

of ext,remal

rays on manifolds

Theorem 7.2.7 (Wibniewski (1991b)). Let X be a nonsingular n-fold. Assume that on X th,ere exists k differen,t eztremal nil,rn2,. . . , rnk: be the dimenkons of the images of contractions Then k(n 1x1

If, furthermore,

- m,)

5 n,.

projective Tayls, and let of th,ese ra:ys.

Chapter 7. Fano Variet,ies with p > 2

146

&(il

- m,,) = 11

or

then p(X)

< k or p(X)

Corollary tirue extremal

~(r-)

=n-

1:

i=l

z=l

5 k + 1, respectively.

7.2.8. If th,e Funo man,ifold rays, then p(X) < dim X.

X contains

only numerically

effec-

The following proposit,ion generalizes 7.1.19, 3.5.3. Proposition o,f a nonsingular subvariety Z c divisor. Assume

7.2.9 (WiSewski

(1991b)). Let p: X -+ Y be the blow-up projective munifold of dimension n > 3 ulong a nonsingular Y of codimension > 2: and let E c X be th,e exceptional that X is u Funo manifold, and Y is not Funo. Then

(i) there exists

an extremul ray Rx on X n,ot con,tmcted by the morphism P such, thut R,x E = 0 (in particular, the exceptional 1ocw.s of R,x is contained in, E), (ii) there exists un extremnl ‘my RE on E of length p(RE) 2 2 not co&acted hJ P, is not numerically

(iii) Kz

effective.

If,

furthermore, p(E) = 2 or the contraction > dim Z, then h’ und Z are Funo manifolds.

of R,x h,us a fiber

of dimension,

Proposition 7.2.9 implies, in particular, the following two propositions. Proposition 7.2.10 (WiAniewski (1991b)). Let p: X + Y be the blo,w-up of a nonsingular n-fold, n > 3, along a smooth curue C c Y: md let E c X be the exceptional divisor. If X is a Funo manifold, then Y is ulso a Fano manifold ezcept for the case Nc,,,y = (‘J-l)“(“-l),

CzlP’,

E E IP’ x P-“.

In this situation X udmits another contraction up of Yl ulong Z e IPTLP’ with N,lyl N U(-1) divisor E. Proposition of a nonsingular ssurface S c Y, fosurfold, then Y

7.2.11 (WiBniewski (199lb)). Let p: X -+ Y be the blow-up projective manifold Y of dimension n > 4 along a nonsingulur and let E c X be the exceptional divisor. If X is a Funo is also a Fano fourfold except for the followkg cases:

(i) S = P2, and Nsly +1p-:x

‘p: X ---f Yl which is the blow@ (3(-l) with, the exceptional

is isomorphic either or’ to T&-3) @ q-l)W-4),

to O(-l)f11(““-2),

(ii) 5’ z II”’ xI?l, NL;ly E U(-1, -l)@(7Lp2), and the generators are nurnericully

equivulent

on Y,

or

to O(-2)

8

o,f S z PI x IP’

s7.2. Higll~r-dirrlensional (iii)

Fano Varieties

with /, > 2

147

S /IUS th P, s t ructure of a ruled surface ~1 S + C over a smooth curve C; in this case there exists another blow-up pl: X + Yl of a man$old Yl along a submanifold Z with the same exceptional &*&or E. The man,ifold Yl is a Fano manifold except for the cases n = 4 or S = @ x @, NSIY (y-1, -lp-2),

The following examples 7.2.10, 7.2.11 really occur.

show

that,

the exceptional

cases in Propositions

Examples (Wibnicwski (1991b)). (i) Let the pair (V, N) be one of the following: (I?~, c3(-1)@(“~‘))~ or (p”: N~z), where N~z is one of the bundles -l)@(‘“-“I). Let us consider the described in 7.2.11 (i), or (P’ x p’, 0(-l, manifold Y := lF”(N* (11Ov), let S be t,he section of the projection associated with the natural morphism N* CD0” + Ov. Then the normal bundle of S in Y is isomorphic to N, and Y is not a Fano manifold. However the blow-up of Y along S is a Fano manifold discussed in 7.2.10, 7.2.11 (i) and 7.2.11 (iii). (ii) Let Al bc th e project,ivization of the vector bm~dlc 0(l) @ 0(l) @ 0 over a smooth quadric Q c P4. Then M contains a section Q associa.ted with t,he morphism U(1) @ O(1) ~1 0 + 0. Take a general divisor Y from the linear system l[+HI, where < is a relatively ample divisor on M, and H is the proper transform of a hyperplanr section of Q. Since I< + HI is an ample and base point free linear system, then Y can be chosen nonsingular. Put S := Y n Q. Then S y p’ x P’, and: by blowingup Y along S, we obtain the case 7.2.11 (ii). Note that unlike the three-dimensional case, Fano manifolds of dimension > 4 can have more complicat,ed contractions of extremal rays than just blowups along nonsingular subvarieties. Exa,mples. (i) Let Y = Y,r c p7 be a nonsingular de1 Pezzo fourfold of degree 5 (see 3.3.1), let C be a conic on Y not lying in a plane contained in Y, and let p: X ---f Y be the blow-up of Y along C. The linear system lH* ~ El, of a hyperplane section, and E is the where H* := p*H is the pull-back exceptional divisor, is base point free and determines a morphism p: X + p4, the contraction of an rxtrernal ray. Since p(X) = 2, and X has two ext,remal rays, then X is a Fano fourfold. The morphisrn y: X ---f p4 contracts a divisor D N 2H* - 3E onto a surface S c p4 of degree 4 with an isolated singular point. The restriction 910: D + S is not equidirncnsional: a general fiber of ylu is the proper transform of a line on Y meeting C, but P/D has also a two-dimensional fiber F, the proper transform of a surface p(F) of degree 2 on Y5 c p7 containing C. In this cast p(F) is a singular point of S. conic, let P := (C) be its linear span. (ii) Let C c p4 b e a non-degenerate Let p: X + p4 be the blow-up along C; and let F := p-‘(P) be the proper transform of P. Then curves on F generate the second extremal ray R 011 X. Therefore X is a. Fano fourfold. The contraction p: X + Y of the ray R is

148

Chapter

small: the only non-trivial codimension 2.

7. Fano

fiber

Varieties

with

p > 2

of ‘p is F. So the exceptional

Now, we consider the case of a conic bundle dimensional case, denote by n the discrimina.nt n := (1~ E Y 1 f-‘(y)

locus of y is of

f: X + Y. As in the threelocus of f:

is a singu1a.r

fiber}

and put, &

:= {y E Y / f-‘(?/)

It is easy to see that

n > &,

Proposition 7.2.12 dle. Suppose that X is be any extremal ray on that f *(-KY) R < 0. its contraction is small

is a non-reduced

and codim&

dirnY 2 3, p(Y) = 1, & = 0, index(X) > dimX

Then

Y is also

The example

> 2.

(WiSniewski (1991b)). Let f: X --7‘ Y be a conic buna Fan,0 manifold, and Y is not a Fano manifold. Let R X which is n,ot contracted by the morphism f and .such Then the locus of the ray R is contained irk f -’ (A,), and has only fibers of dirnensiosn < dim &.

Corollary 7.2.13 (WiBniewski (1991b)). where X is a Funo manifold. Suppose that satisfied: (i) (ii) (iii) (iv)

fiber}.

Let one

f:

X --) Y be a conic bundle,

of the following

conditions

is

~ 3.

a Funo manifold. below

shows that, there exists a conic bundle and Y is not a Fano manifold.

f: X +

Y

such

tIllat X is a Fano manifold

Example 7.2.14 (WiBniewski (1991b)). Let Y be the projectivization of the vector bundle 0 CRc3(2)cH2 over p’, and let 7r: Y ---f p” be the projection. The Picard group Pit(Y) is generated by H := 7r*Op:, (1) and the tautologica.1 divisor v. We consider the bundle E := c3y (11 + 2H) gi 0~ (V + H)“12 and its projectivization f: P(E) + Y. One can prove that the complete 1inea.r system l2< - 4f* H 1, where < is the t,autological divisor on P(E) contains a nonsingular divisor X, and then f: X --f Y is a conic bundle. The divisor - Kx = 2, and let H to

on,e

of

the

the case

(1990a)). be the fundamanifolds

from

$7.2. Higher-dimensional Remark. in dimension

Theorem 4.

Fano Varieties

7.2.15 gives us that the Mukai

with p > 2 Conjecture

149

(7.2.1)

We shall give a sketch of the proof of Theorem 7.2.15 following (199Oa)). First, we note that 1.4.5 gives us the following. Lemma 7.2.16 2 with, p > 2. Then

(Wisniewski

(i)

9: X

if a contraction

+

Y

(1989b)). of som,e

of index

fibers threefold,

of dimen-

(iii)

ray

has

only

over a Funo

sion,

(ii)

(Wisniewski

Let X be a Fano fourfold eztrernal

5 1, then cp: X + Y is a @bundle if th,ere exists m eztremal ray R on thvee-dimensional fiber, th,en, X has a threefold, if th,ere exists a surjective morphism p1 x Y, where Y is either a de1 Pezzo

is true

X

such

that

its

contraction

structure

of pl-bundle

from X threefold

onto a curve, or p3.

has

a

over a Fano th,en X

E

Indeed, let R k)e an extrernal ray on X, and let qs: X + Y be its cont,raction. Then the length p(R) of the ray R is divisible by 2. If /d(R) > 4, then it follows frorn 1.4.6 that the dimension of the locus of R is at, least 4, i. e. Y is a point, and p(X) = 1. Therefore t,he length of each extremal ray R on X is equal < 1. Then, by to 2. Suppose that each fiber of cp: X + Y is of dimension Theorem 1.4.5, Y is nonsingular, and there are two cases: a) dimY = 4, and cp: X 4 Y is the blow-up of a nonsingular 2 c Y of codimension 2, conic bundle. b) dim Y = 3, and cp: X + Y is a standard

subvariety

It follows from the formula for the canonical divisor of a blow-up that, in case a) the index of X is equal to 1, i. e. case a) is irnpossiblc. In case b) let F be a fiber of the conic burldle p, We havt H F = -i Kx F = 1, and, since H is ample? then F is irreducible, and H is a section of t,he conic bundle 9: X + Y. This gives us that cp: X + Y is a @-bundle, and it follows from 7.2.13 that Y is a Fano t,hreefold. This proves (i). To prove (ii), we choose a three-dimensional irreducible component F of a fiber of the morphism y: X - Y. There exists an extrernal ray R’ # R on X such that R’ F > 0. Obviously, fibers of its contraction p’: X + Y’ are of dimension < 1. Thus (ii) follows from (i). Now let $:X + C be a surjective rnorphism onto a curve, and let R I-je an extremal ray which is not contracted by the morphisrn $. We may assume that 21, ha,s cormected fibers, and since h’(0,) = 0, then C = pl. As in (ii), we get that fibers of the contraction ‘p: X - Y of t,he ray R do not intersect fibers of $, and this is why fibers of cp are of dimension 5 1. We get a finite morphism cp x $: X + Y x p’ ( which actually is an isomorphisrn. This proves (iii). By (Prokhorov (1994a)), tl rere exists a nonsingular irreducible divisor H E I-;Kxi. Byth e a d’junction formula and the Lefschetz theorem on hyperplane sections, H is a Fano threefold of index 1 with p(H) = p(X) > 2. ILIoreover,

Chapter

150

Pit(X) NE(H)

= Pit(H), r” NE(X).

and

7. Fano Varieties

N,(X)

= Nl(H).

with p > 2 F ram

Lemrna

3.5.5

we get, that

Lemma 7.2.17 (W’ Gniewski (1990a))). Notation as in Th,eorem 7.2.15. Let pff: H + Y be the restriction morphism. Assume that one of the ,following condition,.5 is satisfied: (i) dim(Y) < 2 (ii) pfj is a contraction of an extremal rag R, in NE(H) and this my considered as an element ofNE(X) z NE(H) zs n~umerically eflective. Then

pff can he extended

to a morphism

cp: X

---f Y.

The idea of the proof is the following. Since NE(X) N NE(H), then the mm~erically effective divisor LH on H determining the rnorphism PH: H + Y can be extended to a numerically effective divisor L on X. Therefore by the Base Point Fret Theorern 1.3.6, the linear system mL deterrnines a morphism y:X+Y. Corollary curve,

then

7.2.18 X

E P1

(Wibniewski x Y,

where

(1989b)). Y

is either

If H has a de1 Pezzo

a morphism

threefold

onto

a

or ps.

Further, we may assume that H has no morphisms onto a curve. We also shall assume tha.t X is not a pl-bundle over a nonsingular threefold (threefolds having structures of pl-bundles are classified in Theorem 7.2.6). We shall consider the following cases: Case p(X) 2 3. We prove that in this case X has a structure of a @bundle. By our assumptions and Corollary 7.2.18, the divisor H has no morphisms onto a curve. Using the classification of Fa,no threefolds with p 2 2, we obtain that p(H) = p(X) = 3, and H has a conic bundle struct,ure 9~: H -7‘ p”. By Lemma 7.2.17, the morphism (PH can be extended to a morphisrn cp: X + p2 which is the contraction of an extremal face of dimension 2. Each of the two cxtremal rays R, R’ contained in this face deterrnines a commutative diagram

where 4: X + Y is the contract,ion of the ray R (or R’). In this case 4 is a @-bundle. Indeed, for this it is sufficient to prove that q!~has no twodirnensional fibers (see 7.2.16). But if 4 has a two-dimensional fiber F, then F is an irreducible component of a two-dimensional fiber F’ of ‘p. But, the fiber F’ can be reducible only in the case when F’ = F + F” and it is a reducible yuadric surface in p’. Since 4 contracts F, then it, also contra& the intersection F n F”, and, by the property of extrernal rays, 4 contracts

37.2. Higher-dimensional

Fano Varieties

the whole F” e P2. But then there exists dim Y = 2, and since fibers of the morphisms an isomorphism, i. e. p = 4. This contradicts

with

p> 2

151

a fiber of $ which is a point, so ‘p and 4 are connected, then $I is the fact that p(X) > p(IP’) + 1.

Case p(X) = 2 and an extremal ray of divisorial type on X. Lrt R be a such a. ray, let 9: X + Y br its contraction, and let E c X he an (irreducible) exceptional divisor of the morphisrn cp. By Theorem 1.4.5, the dimension of each non-trivial fihcr of the morphism cp is al least 2. Therefore dimy(E) < 1. On tl le other hand, if dimp(E) = 0, then X has a st,ructure of PI-bundle (see 7.2.16). This is why p(E) is an irreducible curve C, and the dimension of each fiber of cplf;: E + C is equal to 2. Considering the restrictions ~IH: H 4 p(H), one can prove that cp(H) is a nonsingular variety, and ~IH is the blow-up of p(H) along a smooth curve C. For fibers F of the morphism PIE: E + C, there exists only the possibility F c” P”, and the restriction of 0~ (E) t,o F is isomorphic to 01p~ (-1). By the property of monoidal transformations, Y is a nonsingular variety, and 9: X + Y is the blow-up of Y along a smooth curve C. Further, Y and p(H) are Fano manifolds with p = 1. For the index r.’ of the Fano threefold p(H), we have from 7.1 that 7.’ > 2, and then, by the adjunction formula, we get, that the index of Y is at least 3. But 2H = Kx = p*h’y + 2E, this is why the index of Y is divisible by 2. Therefore the index of Y is equal to 4, and Y = Q c P” is a smoot,h quadric. Whence we get, manifolds 10, 12, 13, 15 frorn Table 12.7. Case p(X) = 2 and numerically effective extremal rays RI, &. Let pi: H + Yl and cpi: H + Y, be the cont,ractions of the corresponding extrernal rays on the Fano t,hreefold X. Then by Lemma 7.2.17, these rnorphisrns can be extended to rnorphisms cp, : X + Y;, i = 1,2, which are also contractions of extrrmal rays. We may assume tha,t 2 < dirnY1 5 dimY2 5 3. Consider for example the case dim Yl = dim Y2 = 2. Then Yl y Yl r” P2, and we get a finite morphism p := (~1 x cpz: X + lP2 x P”. By the Hurwit,z formula, X is a Fano fourfold of index 2 only if the ramification divisor is of bidegree (2,2): i. e. we have No. 4 frorn Table 12.7. Using more subtle arguments, see one can show that if dim(Y1) = 2 and dim(Y2) = 3, (WiSniewski (1989b)) then Yl = lP2, and Y2 = P” or Y2 e Q c IP’ (a smoot,h three-dimensional quadric), and if dini = dim(Y) = 3, then Yl = Y2 z P”. In all the cases the rnorphism ‘p: X + P2 x IP2 is an embedding, and we get Fano fourfolds No. 5, 8, 7: respectively. This finishes the proof of Theorem 7.2.15. We give the following two theorems about the classification of Fano manwit,hout proofs. The proofs use some result,s about, folds X of index i dim(X) contractions of extremal rays. Theorem mnn,ifold of the following

7.2.19 (WiSniewski (1993)). Let X be an n-dimensional r. Assum,e that n = 2r, r > 3 und p(X) 2 2. Then holds:

in&x

Funo one of

152

Chapter

(i) X has a projective sion r + 1, (ii) X has a quudric there

exists

isomorphic

dim(Y) Iiii)

on X

7. Fano Varieties

space bundle bundle

a morphism to a (possibly

structure

structure 4

p: X

with ,o > 2

Y

over u Fano manifold

over such,

degenerate)

of dimen-

manifold Y (1;. e. fiber ‘p-‘(y), y t Y is in P” where k := dim(X) -

a nonsingular th,at each

qwadric

+ l), there

exist

exuctly

two

extremal

rays;

one

of them

is an

extremal

ray

of divisorial

Iiv)

type (see 1.3.8), and the other is numerically effective, and its contraction cp: X + Y deterrnines a structure of non-equidimenszonal bundle of projective spaces, i. e. a general fiber cpP’(:y), y E Y, is isomorphic to IP”, where k := dim(X) - dim(Y), and there exists an ample divisor L on X such that its restriction to pP1(:y) is isomorphic to C&F(~), on X there exist exactly two extremul rays, and their contractions determine on X kwo structures of non-equidimensionul bundle of prqjective spaces. In case (ii) there

(ii)

is a refinement

of Theorem

Theorem 7.2.20 (Wisniewski (1993)). of Theorem 7.2.19. Then X and Y are

(i) Y ‘v (ii) Y E a) b) c) d)

Let X and Y be the same of the following

Corollary. Let X Then p(X) < 2 except manifolds, Conjecture

as in

one

X F Q’ x Q”, where Q’ c IF+’ and for X one has the following X is u divisor of bidegree (1: 2) on P’ X is u divisor of bidegree (1,1) on P” X is a double cover of P” x P’ ramified X is a blow-up of a smooth quadric linear section II?” n Q2’. Q”, P”,

7.2.19.

is u smooth

possibilities: x IPi, x Q1-+‘, in a divisor QS7. c Pzrfl

quadric,

of bidegree along

(2,2),

a smooth

be an 2r-dimensionSal Funo man,ifold of index r > 3. for the case X z IP2 x P2 x P”. In particular, for such ‘7.2.1

is true.

In conclusion we remark that one carmot expect a reasonable classificat,ion of Fano manifolds of dimension > 4 with p 2 2. Refer, for example, to the works (Batyrev (1984), Batyrev (1998)) m which the complete classification of all toric Fano fourfolds was obtained.

38.1. Intermediate

Rationality

Jacobian

and Pryrn Varieties

Chapter 8 Questions for Fano Varieties

58.1. Interrnediate

Jacobian

and Pryrn

153

I

Varieties

It is well-known that two-dimensional Fano varieties, that is, de1 Pezzo surfaces, are rational over any algebraically closed field. For three-dimensional varieties this is far from being true, although it was not easy to prove the non-rationality of some t,hree-dimensional Fano varieties. The first statements about the non-rationality of a srnooth three-dimensional cubic, a smooth three-dirnensional quartic: Fano varieties Vg, Vs and others were made by the classical Italian georneters G. Fano and F. Enriques. But their proofs were not rigorous enough. The first rigorous proofs appeared only in 1971: the non-rationality of a smooth three-dimensional cubic was proved in ClernensGrifliths (1972), of a smooth three-dimensional quartic in Iskovskikh-Manin (1971)) and of some three-dirnensional conic buxldles in Artin-Mumford (1972). Three different methods were used: the method of intermediate Jacobians in Clemens-Griffiths (1972), the classical method of untwisting birational maps based on the NoetherrFano inequalities in Iskovskikh-Manin (1971), and the method conriected with the birational invariance of the BrauerGrothendieck group in Artin-Mumford (1972). La,ter, by developing these methods, proofs of the non-rationalit,y were obtained for many other types of Fano varieties and conic bundles (see Beauville (1977a). Iskovskikh (197910)) Tikhomirov (1986), Pukhlikov (1987), Pukhlikov (1988a), Pukhlikov (1988b), Pukhlikov (1995)) Pukhlikov (1997)) Pukhlikov (1998)) Sarkisov (1980)) Sarkisov (1982)) Endryushka (1984)) Picco Botta-Verra (1983), Conte-Murre (1977), ColliotTheleneeOjanguren (1989)) Saltman (1984), Shafarevich (1990), DolgachevGross (1994), Peyre (1993), Kollar (1995), Kollar (1996b)). We note that the GriffithsClemens method of intermediate Jacobians works only in dimension 3, whereas the other two methods are used in a,ny dimension (see, for example, Pukhlikov (1987), Pukhlikov (1988b), Pukhlikov (1997), Sarkisov (1982)) Colliot-Thelkneeojanguren (1989)) Saltman (1984), Shafarevich (1990)) Kollar (1995): Kollar (1996b)). We discuss briefly each of them. We begin with the method of int,ermediate Jacobians due to Ph. Griffiths and C. Clemens. Let X be an irreducible nonsingular projective three-dimensional variety over @. Assume that h,‘(X, 0~) = h”(X,Ux) = 0, and let H”.“(X) @ H2,1(X) @ Hi.‘(X) 63 Ha,“(X) be the Hodge decomposition of the three-dimensional cohornology of X. The condition h”(X, 0~) = 0 is equivalent to H”.“(X) = H”.3(X) = 0. Therefore the imegration of (2, l)-forms over 3-dimensional cycles determines an ernbedding Hs(X, Z)/(torsion) Lf H2.1(X)* as a full lattice of rank 2h2.1 into the complex vector space iY’,‘(X)*. The alternating integral-valued intersection form E on H3(X, Z) is unimodular by Poincare duality, and the corresponding Her-

Chapter

154

8.

R&ionality

Questions

for Fano Varieties

mitian form H on H2.r(X) * is positive definite. Therefore, Frobenius criterion, the complex torus g(X) := H’.‘(X)*/ a principally polarized abelian variet)y with the polarization Definition the intermediate

8.1.1. In the preceding not&on, ,Jacohian of the uuriety X.

the pair

Applications of intermediat,e Jacobians in birat,ional the following observat,ion of Clemens-Griffiths (1972) Tyurin (1979), Cilibert,o (1980)). Lemma (i)

by the Riemann im(H’(X, Z)) divisor 8. (J(X),

0)

is

zs culled

geometry are based on (see also Tyurin (1972) j

8.1.2.

Let (T: X’ + X be the blowup of (I, nonsingulur (th,ut is: C is either a curve or a point). Then

J(x’)

(ii)

I

J-(X) = { J(X)

> @J(C)7

in-educible

center

C

c X

if C is u point; if C is a curve

where J(C) is the Jacobian of the cu’1‘11~ C with the principal polarization determined by the Poincare’ divisor 0. Every p ram1 “p a 11y polwized a,beliun variety (A. Z) can be decomposed in a unique way into a direct sum, of principull:y polarized simple ubeliun uurieties

(A, E) = zil(AiT

ET,).

Tkris lemma and the theorem on resolution of indeterminacies of rational maps immediately imply the birational invariance of tkrose components (the so-called Griffiths components) from tlrr: decomposition (g(X), @) = c$ (A,, , 0;) into a sum of simple components whiclr are not Jacohians of /=I curves. In other words, if one denotes by J g - 4. (ii) (the T ore 11’1 theorem, see, for example, Griffiths-Harris (1978)). The Guuss mup G: 0 4 IF1 is finite on the set of nonsingular points 0” = 0 Sing 0, a& the rurn~~cuk~on divisor B c IF”-’ is the (projective) dual of the kitiul curve C. Thsus, the principully polarized variety (J(C), 0) uniquely determines the curve C. Intermediat,e Jacobians of t,hree-dirrlcnsiorial varieties often turn out to be so called Prym varieties. (this is true, for example, for varieties whicah can be represented as conic bundles over rat’ional surfa.ces, see below). These varicties were studied first in papers of F. Prym, W. Wirtinger, F. Schottky, H. Jung, and others in connection with the study of double covers of curves and the Schottky problem of description of the moduli of Ja.cobians of curves in the moduli space of principally polarized a,belian variet,ies (see the survey and also Clernens (1974)). of Beauville (1977b), D. Mumford was the first who drew attention to the possibility of applications of Prym varieties t,o the birational gcornetry of three-dimensional varieties in the appendix to the paper of Clemens-GrifMhs (1972). He also studied the ca.se of unramified (and also ramified in two points) double covers C + C of nonsingular curves C from the point of view of distinguishing their Prym varieties Pr(c: C) frorn .Jacobians of curves (Mumford (1974)). Mumford’s rcsult,s were extended t,o singular curves with normal crossings (such a situation arises naturally in the study of conic bundles, see below) by Bcauville (1977a), Bcauville (1977b) and Shokurov (1983), Shokurov (1981). Definition 8.1.4. Let (c, i) be the pair consisting of u complete reduced (possibly redu cz _‘bl e) curw: C- with at most ordinur:y double points und an involution i on c (that is, i‘’ = id) operating non-tri~uiully on every *irreducible component of the curve cl. Den,ote by C the quotient e/i, and by rr: c ---f C th,e correspon,ding double cover. The involution i induces un inwolution %* on the group Pic(@, and T induces the norm, map Nrn: Pit(c) 4 Pit(C). The con,netted commutative algebraic group Pr(c, C) := ker(Nm)” = ker(id +i*)” = (id -i*)g(C) 2.s culled the (generalized) Prym wwiety of the pair (I??, i) (the symbol “ ’ ” daotes the identity componernt. The notion eties (being

of polarization can be easily extended to generalized Pryrn varialgebraic groups, they are not necessarily abelian variet,ies when

156

Chapter

8. Rationality

Questions

for Fano Varieties

I

C is singular). Under certain conditions, Pr(C, C) is an abelian variety the polarization of which is divisible by 2 a.nd after the division becomes principal. This is true, for example, if the following Beauville conditions are satisfied (Beauville (1977b), Shokurov (1983)): (B)

Singe = {:r E C / i(x) = z}, and ?r(Sing C) = Sing C, that is, the fixed points of the involution i are exactly all the singular points of the curve C, and the involution i preserves (that is, does not interchange) the two branches of the curve in arl_y singular point. If C is nonsingular, then C is also nonsingular, and 7r: C + C is an unramified double cover.

For the following theorern, see Murnford Shokurov (1983)) Debarre (1989a).

(1974),

Theorem 8.1.5. Let (C, i) be the pasir consisting - 1, g # 5, and an involution i on C satisfying following addltionnl con,dition: 29

(*)

for ewery non-trivial

Let C Prym Then in the (i) (ii) (iii) (iv)

decomposition

C = Cl U&,

(1977b),

of a curue C of genus conditions (B) and the we have #(Cl

nCz)

> 4.

= C/i be the quotient c?Lrue, und let P = Pr(C, C) be the corresponding variety (pa(C) = g, dimP = g - 1) with th,e polarization divisor E. (P, Z) is the Jacobian of a curve or a sum, of Jacobians of curves only following cases:

C 8s a hyperelliptic curve; C is a hyp ere 11’~,pt’ac curve with two points glued together; C is a trigonal curve: C is a double cover of a plune curve of degree 5 corresponding theta-characteristic.

Here we mean by a h,yperelliptic (respectively a finite morphism of degree 2 (respectively From (**)

Beauville

conditions

(B) one can easily

for every decomposition

to an even

trigon,al) curve a curue 3) onto ltu’.

deduce

the following

possessing

condition:

C = Ci U Cz we have #(Cl

n C2) 55 O(mod2)

If Ci nCZ = {p. q}, and z”,‘, i = 1,2, are curves obtained from C, by identifying the points p and y, then P = PI x Pz, where P,, are Prym varieties for 2;,’ with the induced involution (see Shokurov (1983)). This is the reason for the condition (*) to be included in the hypothesis of the theorern. Prym varieties arc used in three-dimensional birational geometry mainly in the following situation. Let f: X + S be a standard conic bundle over a nonsingular projective rational surface, that is, f is a surject,ive morphism every fiber of which is isomorphic to a conic in p2 (perhaps degenerate), and for every irreducible curve r c S, the surface XT := f-‘(r) is irreducible

$8.1. Intermediate

Jacobian

and Prym Varieties

157

(this is equivalent to Pit(X) = f* Pit(S) + Z). Let C c 5’ be the discriminantj curve of f: and let C be the curve parametrizing irreducible components of degencratc corks over points from C. Then if C # 0, it iz a reduced clu-ve with norma. crossings, and f induces a double cover 7r: C + C sat,isfying conditions (B) (see Beauville (1977a)). For the following theorem, see Brauvillr (1977a), Beltrarnetti (1985), Beauville (1989). Theorem surface

8.1.6.

S with

J(X)

the

is isomorphic

C is the

curve

Let f: X 4 S he u stundard conic bundle over u mtionml curve C c S. Then the intermediate Jucobiun to th,e principully polurized Prym, variety Pr(c, C), where

discriminant

parametrizing

components

of degenemte

tonics

over

points

from

C.

A. Beauville considered the case S = P2 and obtained as a consequence of Theorems 8.1.5, 8.1.6 the following statement: if S = F”, and degC 2 6, then the variety X is non-rational since in this case Pr(C, C) is neither t)he Jacobian of a curve nor a sum of Jacobians of curves. Applying this method and using t,he ckssification a.nd Theorems 8.1.5, 8.1.6, we get the following results for three-dimensional Fano varieties. Theorem 8.1.7. Every nonsingular three-dimensional Fano varieky X of one of the types listed below is birution,ull:y isomorphic to a standard conic bundle, and its intermediate .Jacobian (bein,g a Prym uaviet:y) is diRerent from Jacobians of curves. In particular, X is not rational (see Clemens-Grifiths (1972), Bea,uoille (1977a,), Tyurin (1979), Clemens (1974), Shokurov (19&Y)): (1) X 2 X3 is a three-dimen.sional cubic, p(X) = 1 (Table 12.8, No. 13); (2) X z X8 is a com,plete intersection of th,ree quadrics, p(X) = 1 (Table 12.2: No. 4); (3) X z Xl4 is u section of the Grussmunnian Gr(2,6) c PI” by a, subspace of codimension 5, p(X) = 1 (Table 12.2, No. 7); (4) X + p1 x P2 is a double cover with the ramification divisor of bidegree (2,4), p(X) = 2 (Table 12.3, No. 2); (5) X is the bl ow-up of X3 with center in a plane cubic curve, p(X) = 2 (Table 12.3, No. 5); (6) X c p2 x P2 is a divisor of bidegree (2,2), p(X) = 2 (Table 12.3: No. 6); (7) X is a double cover of a Fan,o variety X7 c p8 ramified in a divisor p(X) = 2 (Table i2.3, No. 8); D N -Kx7, (8) X is the blow-up of XL< with center in a line, p(X) = 2 (Table 12.3, No. 11); (9) X is u double cover of p1 x p1 x @ *rumijied in a divisor of multidegree (2,2, a), p(X) = 3 (Table 12.4, NO. I). Moreover, type

(31,

Tregub

every and

(1985),

nonsingulur

versa Tukeuchi

vice

variety of type (1) zs b’zru t’zonul to sorae uurieby of (see Fano (1930), Iskovskikh (1979b), Tregub (1990), (1989), and also Chap. 4).

158

Chapter

8. R.ationality

Questions

for Fano Varieties

I

Sketch sf the proof. The projection from a line represents X3 as a standard conic bundle over p” with the-discriminant curve C C p2 of degree 5. The corresponding double cover ?r: C + C is associated with an even thetacharacterist,ic (see, for example, Tyurin (1972), Beauville (1982a), Beauville (1977a)). Hence, by Theorem 8.1.5 (iv), Pr(e,C) is not, the Jacobian of a curve. By Theorem 8.1.6, g(X,) e Pr(??,C): and therefore X3 is not rational. Likewise, by Theorern 4.3.4 (ii), every smooth variet,y Xs is birat,ional to a standard conic bundle over p2 with the discrirninant curve C c p2 of degree 7 and therefore is not rational by virtue of Beauville’s result (Reauvillr (1977a)). Varieties of type (3) are birationally isomorphic to varieties of type (1). This was first noticed by Fano (1930) ( an d wa,s re-proved in Iskovskikh (19791))). Ot,her variants of the proof were suggested by Tregub (1985), Tregub (1990) and Takeuchi (1989). By Theorem 4.3.4 (iv), every nonsingular X14 is birational to a standard conic: bundle over p2 with t,he discriminant curve C c p2 of degree 5. How to transform this conic bundle into the conic bundle obtained from a cubic X:{ by projecting from a line was shown by Tregub (1990), see also Chap. 4. Varieties of types (5) and (8) are obviously birational to X:3; and va,rieties of type (8) are staildard conic bundles over p2. Moreover; varieties of types (4)-(7) are also standard conic bundles with respect to the nat,ural projections:

(4): (5): (6): (7):

X X X X

+ 4 t 4

p2 p2 p2 l?’

wit,h with with x p’

t,he degeneration curve of degree 8: the degeneration curve of degree 6; the degeneration curve of degree 6; with t,he degeneration curve of bidegree

(4.4).

Prym varieties are often used to prove the non-rationality of many types of Fano varieties which are birationally equivalent, to conic bundles, in part,icular, singular Fano va.rieties. We note some of these results. Proposition

8.1.8.

(i) Let X,, C P” he a hypersuflrfuce of degree m > 4 in P4 which bus an (m - 2)-fold line and has no other singularities. Then the projection from this line represents X,,, as u conic b~undle over P2, and X,,, is n,ot rationd (see Conte-Mur-re (1977)). In particular, if m = 4, then X,,, is a Fano TJariety ~wdh cunonicu1 singularities. (ii) A gen,eral three-dimensiond Enriques variety X c P4 defined by 0~71, equation of th,e form

$3.1.

Intermediate

Jacobian

and

Prym

Varieties

159

l;s birationall:y equivalent to a stun&d conic bmdle over a rutionul surTh,is mriety is unimtionul bsut not mtionul by Endryushka (1984), Picco Bottu-Verru (1983), which conforms with the clussicul statement due to F. Enriqv,es ubowt this.

face.

We remark that Pryrn varieties are used in Beauville et al. (1985) to prove the non-rationality of some three-dimensional conic bundle X such t,hat X xp3 is rational. This gives a counterexample to the well-known &rational) Zariski problem. 8.1.9. A. Beauville (1977a) used Prym varieties to prove the nonrationality of general Fano varieties of some types by applying the so-called Clemens degeneration method (Clemens (1974)). The idea of the method consists in the following. Let 7r: X + S be a family of three-dimensional projective varieties over a smooth (not necessarily complete) curve S with nonsingular X such t,hat,: X, = rTT1 (s) is nonsingular. (1) for every s E S. s # SO, the variety h”(X,, 0,) = 0 for i > 1, and H’I(X,$: IE) has no torsion; (2) X(1 = T-1 (so) has at most ordina.ry double singularities. Let A, be the moduli of principally pola,rized abelian varieties. Then by virtue of the results due to Zucker (1976), there exists a morphism j: S - sg + A, sending every point s E S - sg to the intermediate Jacobian ,7(X,?) of the three-dimensional va.riety X,q. Moreover, this morphism can be extended to a morphism 5: S --f As, where A, = A, U A,-1 U . . . U A0 is the Satakr compactificat’ion, so that, I = J(XA) with 3(X6) being the intermediate Jacobian of a nonsingular resolution X,” of X0. Next, one uses the result of W. Hoyt that states that the closure M, c A, of the moduli M, of curves of genus 9 under the natural ernbedding M, c A, consists of Jacobians and products of .Jacobians of curves of genus < 9. Therefore if the sum of the Griffiths components Jc;(X[,) is non-trivial, then there exists an open neighborhood U c S, sg t U, such that JG(X,?) # 0 for s E U. Thus t,he general variety X frorn the family 7r: X + S is non-rationa,l. Beauville (1977a) and Tyurin (1979) considered some families of Fano varieties for which XE, is a conic bundle over a rational surface, and 3(X{)) is represented by a Prym va.riety different from Jacobians of curves and their

160

Chapter

X. Rationality

Questions

products. In this way the non-rationality three-dimensional Fano varieties:

for Fano Vari&ies

is proved

I

for t)lie following

general

a double cover of p’ ramified in a sextic (Table 12.2, No. 1); a double Veronese cone (Table 12.2, No. 11); a double cover of p’ ramified in a quartic (Table 12.2, No. 12); a quartic in p4 (Table 12.2, No. 2); an intersection of a quartic and a cubic in Y” (Table 12.2, No. 3); a section of the Grassmarmian Gr(2,5) (Table 12.2, No. l), a.nd ot,hers. In some cases one uses the continuation of the morphism j for worse drgenerations than with ordinary double points: for example, the quartic X0 has a double line of singularities. The more delicate degeneration rnethod is used in Collino (197913) to get, a simple proof of the non-rationality of a general cubic, and in Bardelli (1984) to proof the non-rationality of genera.1 three-dimensional Fa,no varieties of some types and of some general three-dimensional varieties with a pencil of de1 Pezzo surfaces, for example, of a general hypersurface X c l?‘i x p” of bidegree (7), 3) for n > 2. The idea of Collino (1979b) consist,s in t,he following: the limit cubic X0 has one ordinary double point,, and 3(X;) is of course t,he ,Jacobian of a curve of degree 4 with the natural polarization. so the previous method does not work. For a family of cubits 7r: X + S (as above), one considers, following Zucker (1976)) the associated family p: A 4 S of algebraic groups polarized by a relative Cartier divisor D. The group A,s = p-‘(s) is the intermediate Jacobian 3(X,?) of the nonsingular cubic X,s c IY’ for s # 0. The group A0 = p~‘(s~r) is an extension of the Jacobian of a curve 3(X;) by the one-tlimcrlsiorlal torus T:

The divisor D induces a polarization on A,) coinciding on z(X(r) with the nat,ural Poincare polarizat,ion. So the pair (AC), Do) is considered as the interrnediate Jacobian of the singular cubic X0. Collino (1979b) proved t,hat (AC), Do) is isomorphic neit,her to the generalized (polarized) Jacobian of any curve nor to the product of generalized Jacobians of curves (with ordinary double points). This irnrncdiately irnplies the non-rationality of the general cubic X c p”. Bardelli (1984) considered families of t,hree-dimensional varieties similar to the above ones but with the special fiber X0 being a reduced divisor with norrnal crossings consisting of at most three nonsingular irreducible components X6, i = 1,2,3, and such t,hat Hi (Xl, n Xi, Z) = 0 for % # j. Aga.in following S. Zucker, one introduces a farnily of intermediate Jacobians p: A + S. The generalized intermediate .Jac-ohian A0 = 3(X,,) = pm1 (so) is defined in t,erms of polarized mixed Hodge st,ructures: it, is an ext,ension of the polarized abelian variety nf=, 3(X;) w h’ ic 11 is the product, of t,he interrnediate .Jacobians of nonsingular components by some torus T. The torus T can be

$8.1. Intermediate

Jacobian

and Prym Varieties

161

polarized in an appropriate way using the local Picard-Lefschetz transformation Hs(X,~, Z) + H:s(X,~, Z) around t,he point so E S. Assuming that every fiber X,s? s E S - so, is a. rational variety, Bardelli (1984) proved that the corresponding farnily p: A + S over some neighborhood 1J c S, CJ3 so, is a. polarized family of Jacobians of some degenerate family of curves X: C + U wit,h C being nonsingular, X-l (so) being semistablr, and X-l(s) being nonsingular for s # SO. In particular, under these assumptions, the group A0 = J(X,,) should be t.he generalized Jacobian of a semistable curve Co wit,h the corresponding polarization. The non-rationalit,y of general three-dimensional varieties of some types can now be proved by const,ructing families with X0 such that, the intermediate Jacobian J(Xo) is not isomorphic t,o the generalized Jacobian of any semistable curve. This is very useful when studying t,hose three-dimensional varieties which it is difficult, (or impossible) to deform to a conic bundle. In t,he next, chapter we shall see that the drgrnerat,ion method can be successfully used to study properties of the Abel-Jacobi map. 8.1.10. Applications of Prym varieties in birational geometry became an incentive to their cornprehensive study (see survey Beauville (1989)). Since Prym varieties are the simplest generalizations of Jacobians among all principally polarized abelian varieties, one poses and study for them the same questions as for Jacobians: analogues of the Torelli theorem, the Schottky problem, the geornetry of the polarization divisor, etc. Let A, be the rnoduli of principally polarized abelian varieties of dimension p, and let zp c A, be the subspace of Jacobians of curves. Let ‘P,, c A, be the subspace of Prym varieties of unramified double covers (: + C of curves of genus p + 1, and let Pp c AI1 be its closure. According to the classical results due to W. Wirtinger, Jp c P?, is an irreducible subvariety in A, of dimension 3p for p > 5, and Pl, = A, for p < 5. Let R, be the moduli of unramified double covers r: c + C with g(C) = 9; in other words, this is t,he moduli of curves C, E M, with a non-zero theta.characteristic. It is a finite cover of degree 2’” - 1 of the rnoduli space M,q. The correspondence (G, C) -* Pr(%;, C) determines the Prym map Pr,: R,+l + A,. Ry the preceding staternent about dimensions, this rnap is finite in the general point for p 2 5 because curves of genus 9 = p + 1 depends on 3p parameters. By analogy with the period map (or the Torelli map) 3: M, + A, associating a curve with its Jacobian, the natural question arises about, the injectivity of the map Pr. The following analogue of the Torelli theorem is true for Prym varieties: the Prym map Pr,: R,+l --j A, is injective in the general point (that is, is birational onto its image) for p > 5. For p > 8 this was first proved by Kanev (1982)) and for p 2 6 by Picco Botta-Vcrra (1980). A more geometric proof was obtained by Welters (1987) and Debarre (1989a). The tetragonal construction of R. Donagi shows, nevertheless, that the Prym rnap is not, injective for any p. For a more d&ailed study of the map Pr and other questions related to Prym

162

Chapter

8.

Rationality

Questions

varieties (in particular, an analogue survey by Beauville (1989).

s8.2. Intermediate

for

Fano

Varieties

of the Schottky

,Jacobian:

the Abel

problem),

-Jacobi

I

refer

to the

Map

By an analogy wit’11 the classical Abel-Jacobi map Pit”(C) + J(C) for a curve C, one can define a hornornorphism cp: A”(X) --f J 3. They first construct exarnples of unirat,ional but not rational three-dimensional varieties over @ among conic bundles over rational surfaces 7~ X + S with Tors H”(X, Z) # 0. Then using product,s of the type X x P”: 7~ > 0, they give examples of nonsingular unirational non-rational varieties in any dimension > 3. The Artinnlumford construction is based on the following calculation of the cohomological Brauer group Br(X) f or standard conic bundles over a smooth projective surface 5’. Theorem 8.3.1 (Artin-Mumford surface. Th,en there ezists the following

(1972)). exact

Let

S

sequence

be a smooth of grosups:

projective

$8.3.

7h3-e

The

/LC’ = up;’

Brauer

Group

= u Hom(IL,,:

as a Birational

Q/Z),

ur~d

pIL

167

Invariant

denotes

the

group

of n-th,

of unity, $k(S)) k the classical Bmuer group of the jbnnction field k(S) over S, and the homornorp}aasrn (Y sends Q class of algebras frurr~ Br( k(S)) to the set of its local invaraants from the g*roups H&(C,, Q/Z). roots

Example 8.3.2. (see Sarkisov (1982), Iskovskikh (1987)). Let 7r: X --f S be a standard conic bundle over a nonsingular rational surface S with the discriminant curve C = C Ci, where C,, are irreducible components. It is known that CJ>s a reduced curve with normal crossings, and the corresponding double cover C + C satisfies the Beauville conditions (see 8.1.4). The group Br(S) equals zero because S is a rational surface, and the cohomological Brauer group is a birational invariant (trivial for projective spaces). To the general corresponds a fiber X,, = ~-l(~j) (,rI E S is the generic point) there uniquely quaternion algebra A,,. Let, [A,] E Br(k(S)) b e i t s c1ass. Then the homomorphism Q from the ArtirlPMumford exact sequence is injective. and an element by it,s local invariant,s. of order 2 of the group Br( k(S)) is uriiquely determined Local invariants in this case are exact,ly double covers ii,: e,, + C,, which arc defined by components of fibers over C,i of the morphism 7r, and [3(x) =

2

i k

if ?;- 4 C is ramified m 6ther ;Bs,,. : L ,LY

at the point, 11:t C,;

The surjectivity of y means t,hat the number of ramification points on every component, is cvcn. is a reduced curve with norrnal crossings on S, Conversely. if C = CC, and ?r: C + C is its double cover satisfying the Bcauville conditions (B) from X.1.4, then using t,he Artin bhnnford exact sequence, we can uniquely recover some element of Br(k(S)) of order 2. If it is represented by some quatcrnion algebra over k(S), then there exists a standard conic bundle with thcsc local invariants. This is the case in our sit,uation. Indeed, according to Merkur’ev (1981), every element of order 2 is a product, of classes of quaternion algebras. But since t,he function field k(S) being of transcendence degree 2 over an algebraically closed field k is of type C2 in the sense of Serre, we have that, t,he class of a product of quaternion algebras can also bc represented by some quat,ernion algebra. Therefore one can const,ruct various standard conic bundles over a. rationa. surface S start,ing wit,h a reduced curve C = C C, with normal crossings and a double cover Z: ?? + C satisfying condition (B). In (Artin-Mumford (1972)) the authors proved the following result which can be used to construct niimerous examples of imirational but, not rat,ional three-dimensional varieties.

168

Chapter

8. Rationality

Questions

for Fano Varieties

I

conic bundle over a rational Theorem 8.3.3. Let X/S h,e a standurd face S with the discriminant curve C. Th,en Br(X) ‘v (2/2Z)“-’ whe,re th,e number of connected componenA~ of the curdle C.

SWc is

This result, shows that to construct c~xamples of non-rational threedimensional varieties, it is suflicirnt to specify local invariants and an unconnected curve C on some rational surfa.ce S. The local invariants can be chosen in such a way that, the resulting conic bundle X/S is unirational. This construct,ion together with the results of Sarkisov (1982) can also be used to produce examples of unirational non-rational conic bundles with trivial int,ermcdiate .Jacobia,n. For nonsingular three-dimensional Fano varieties, the Brauer group is trivial. But for Fano varieties with singularities (to be exact, for their dcsingnlarizations) it can he non-trivial, as, for example, in Artin-Mumford (1972) where the simplest examples of non-rational unirational three-dimensional varieties are given by double covers of P” ramified in singular sext,ics, t,hat is, by three-dimensional Fano varieties with double singularities. Examples of nonrational unirational elliptic threefolds were constract,ed by Dolgachev-Gross (1994). D. Saltman (1984) used the birational invariance of the cohomological Brauer group to const,ruct counterexarnples t,o the E. Noether problem of the rationality of fields of invariant,s for finite linear groups operating on vector spaces over an algebraically closed field. The problem can be forrnulated in geometrical t,erms as follows. let G be a finit,e group operating by linear transformations on a projective space P”. The problem is whether the quoCent X = PI/C 7 is rational. It is clear that X is a unirational Fano variety having in the general case log terminal singularities. To prove the non-rationality of X, it suffices to find a nail-zero element from Br(k(X)) = H’(lc(X),m)*) w h ic.l1 is ‘?mramified” in all the discrete valuations of the function field /C(X) = ,(P’)” on X, t,hat is, t,he field of invariants of the group G. It turns out that this ca.n be done in terms of the group G. and therefore count,erexamples to the E. Noether problem ca.n be constructed. Saltrnan’s arguments from (Saltman (1984)) were simplified by Bogomolov to t,heir interpretation, the bira(1987) and Shafarevich (1990). A ccording tional invariant, the unramified Faddeev-Brauer group @Br(k(X)) (in the notation of Shafarevich (1990)), coincides with some subgroup of the Schur multiplicator H2(G, k*) = H2(G, ~1,) of’ the group G (here 1-1c /c* is the subgroup of all roots of unity) consisting of elements which split when restricted t,o every commutative subgroup H c G. The simplest examples of groups G with non-trivial subgroup @Br k(X) c H,(G, /L) are given by iiorlcorrlrnllt,ativc p-groups of period p where p is a prime. The group @Br(r(X)) does not dcpend on the choice of an exact represent,ation of the group G (see Bogomolov (1987)).

$8.3. The Brauer

Group

as a Birational

Invariant

169

Other examples of urlirational non-rational quadric fibrations of dimcnsion d > 3 were constructed in Colliot-ThC?lPne~Ojanguren (1989), Ojanguren (1990) even with trivial Bra.uer group. They use the higher cohomological Grothendieck invariants and in particula,r the so called unramified Witt group of classes of quadratic forms over the function field k(X). Their arguments were generalized by Peyre (1993) to construct new examples of unirational non-rational function fields. We note finally a very important result, of KollBr (1995). We are speaking of a new rnethod of proving non-rationality of some general hypersurfaces (which are of course Fano varieties)*. Theorem 8.3.4. Let X,1 c P”+l be a general h,ypersurfuce of de,gree d > $ (n + 3) Assume in addition that n and d are even. Th,en X,, is not ruled (in particudar, is not rational). The generality here meaRCs that the assertion is true for euery poin,t of the mod& space outside a countable set of closed subunrieties. The idea, roughly speaking, consists in the following. It is not difficult to check that if r\‘nk contains a positive subsheaf, then t,he variety X is not separably uniruled (see 6.1). But in characteristic zero t,his condition can be satisfied only for varieties of general type. The situation changes if we reduce our variety to characteristic p > 0. The variety Xd can be deformed into a cyclic cover 7r: Y + X of degree p of a hypersurface of degree d/p (it is assurned that p divides d). Further, Y ca,n be embedded into the total space of an invertible sheaf L over X. The natural exact sequence

T*L-p 5 splits

in characteristic

n*L-’

@ 7r*f& +

n& +

0

p > 0:

We obtain an embedding r: An-l (Coker d) q ~‘“-lQh. If Y were nonsingular, then /I”-~ Coker d would be an invertible sheaf with first Chern class equal and therefore would be ample for p = 2; degX = z > to 7r*Icx + p7r*cl(L), ;(n> + 1). But th P variety Y 1la.s singularities, so one should lift the ma.p 7 to a resolution r: Y’ + Y and apply this remark to subsheaves of A+‘nb,. Similar arguments a.re used t,o prove the following result,. Theorem

8.3.5.

Let X,1 c IV+’

be a general

hypersurfuce

over C. Then:

(i) ifd > 2[?1, th,en X,1 is not ruled; (ii) $‘d > 3[yl, then X,1 is not birationully equivalent to a conic bundle; (iii) if d = n + 1, then every domin3nnt r&on,a,l mup Y x P1 - - +X(f with dirnY = n - 1 has degree divisible by every prime less than &i.

* For other applimtions

of this method,

see (Kollk

(19961-j), Kolliir

(1996a)).

Chapter

170

9. Rationality

Questions

for Fano Varieties

II

Chapter 9 Questions for Fano Varieties

R,ationality $9.1. Biratiord

Autornorphisms

II

of Fano Varieties

As was already mentioned at the beginning of the previous chapter, the second way for proving the non-rationality of some types of Fano varieties is to USC the classical method for studying birational maps between Wrieties satisfying the adjunction termination condition due to nl. Noether and G. Fano. The method is based on the Noether-Fano inequality: if x: X - - +X’ is a birational map determined by a linear system M. and if for X’ the adjunction termination condition is satisfied (that is, 1D + nh’x 1= 0 for every divisor D and integer n > no), then (under some cxt,ra COW dition) the linear system 121 should have a. base c:omponent with sufficiently large muhiplicity. As a matt,er of fact, this is the statement about the birational invariance of the adjunction termination condition formallatcd in a suitable way. To be more exact, let x:X- +X’ be a birational map of smooth projective varieties, and let the following t1iagra.m

(91.1)

be a resolution

of indrt,erminacies of the map x by means of blow-ups i = 0, i N ~ 1 with nonsingular crnt,ers R, C X;, where y is a birational morphism. In what follows we shall USC the following notation (the same as in Iskovskikll-hlanin (1971), Danilov (1982)): oi+1.1:

x,+1

--+

XL,

E 1+1= cTI+l.L -’ (&)

X,+l are exceptional divisors, i = 1.. , N - 1; e, is the class of E, in the component AI of the Chow ring A(X,): 0, is the class of B, in A”?+’ (Xi) j where 6; = dim X - dim B, - 1; f f 2.J := (TJ+l.,l 0 ” 0 (T,.,- 1: xi + x,,: i > j; C

3Cl.l. Birational

Cl :=

cT1.0

For the (1971).

0 .

‘0

following

Automorphisrns

of Fano Varieties

171

XN + X.

CJ~,ij-l,

lemma,

refer

t)o Iskovskikh

(1979b),

Iskovskikh-Manin

Lemma 9.1.1 (NoetherPFano ineyualit,y). In the preceding notation,, let x: X - +X’ be CL b%,rutional m,ap of smooth projective wrieties. Let H’ be u numerically effective divisor on X’. Assume thut there exists an integer nzo > 0 such th,ut th,e iknew system lH’ + rn,Kx, 1 is empty (that is, it does n,ot contain an,y cffectke dtiuisoi for e’uery m, > mo. Let H = g*p*H’, und let ~1,. . UN be inte,qers such thut $H’ = a*H - c ~oh,,E, (in the notution of (9.1.1)). Th,en u, > 0 ,for any % = 1,. , N, und for euery m > mo either the linear system IH + rnh’x / on X is empty or th,ere exists a “maximu singulnrity “, that is> B;-l such that vi > rn,fi, , where

(9.1.2)

6, is us ubow.

Proof. The divisor cp* H’ is mmlerically effective as well as the divisor H’. Therefore p*H’ (~h,;,f, __ > 0; where fl is the class of n line in a. fiber of t,he exceptional divisor E,. Hence by the projection formula: 0 5 cp*H’.c~K,,f,

= (0-H

- ~v~o~,~E~,)

.ah,,f% = -v,(e,f%) = u,

The numbers v, can be interpret,ed as mult,iplicities divisor H (if it is effective) along the base components Recall now that h’x, = a*Kx + c SicrJf&E, Consider

the m-th

adjunction

p* H’ + mKXN

= u*(H

and apply

w* to the both

of singularities Bi-1, i = 1,

of the . , N.

on XN: + ml(x)

+ e(rnh.; /=I

sides of this equality.

- v,)o&,~E~ As a result,

(9.1.3)

we get,

N

H’ + rrlKx/

= p,a*(H

+ mKx)

+ yx+

~(lrs& /=l

- ~~)afv,~E,

,

because p*v*H’ = H’. and (PUKE, = Kx, (the morphism cp is birational). By t,he assumption, the left hand side is not equivalent to any effective divisor for m > UL(), and therefore neither is the right hand side. This means that if the linear system IH + n&x / is not ernpty, t,hen Ip,a*(H + mKx)I # 8. Hence the divisor C(rnS, - v,)(T;V,~ Ei is not effective, that, is, there is at, least one index i such that v, > n~6,, which is desired.

172

Chapter

9. Rationality

Questions

for Fano Varieties

II

We remark that this inequality can be refined by considering on the right, hand side of (9.1.3) the proper transforms on X of exceptlional divisors E, instead of their tota,l inverse images ah.L E,,. Drnot,e the proper transforms of E; with respect, to (TN,L by F, = (c~;f,~&(E~). Then ah,;E; = X,27 rJ,F,, where T,]% a,re non-negative integers. The second surnmand on the right hand side of (9.1.3) can be rewritten in the form c,“=, Cjri (m~?~ - vJ) F,. Since the divisors FL are effective, we obt,ain the following strengt,hened Fano inequality. Lemma 9.1.2. system (H + mKx/

Under the conditions of 9.1.1, if for m > mo the linear is not empty, then there exists un index i such that

If for some i the inequality (9.1.2) or (9.1.4) ..is satisfied, we shall call the base component B,-l a maximal singularity of the linear system IHI on X. Remark 9.1.3. In the classical theory, the inequalit,ies (9.1.2) and (9.1.4) are used in the following way. If some birational map X: X - - + X’ of varieties with the adjunction termination condition is not a rnorphism, then, as a rule (for example, for Fano varieties with p = l), the linear system determining the map X has maximal singularities. Next, either the existence of rnaximal singularit,ies leads to a contradiction, and then X is a morphism (usually an isomorphism, as in the case of Fano variet,ies with p = l), or maxirnal singularities satisfy strong restrictions, and with every such singularity one can associate a certain elementary birational rnap “sirnplifying” ( “untwisting”) t,he map X. If, as a result of t,hese simplifications, we a,rrive at the situation when for a given rn > me all maximal singularities are already “untwisted”, then by Lemrna 9.1.2 the 1inea.r system IH + mKx 1 is empty. Further, it turns out that there exists a birational rnorphism (a,n isomorphisrn or even the ideritit,y isomorphism) 41: X + Xi such that the linear system l&(Hr) + ,rnlKXI / is not empt)y for ml < m. Since this process decreases non-negative integers (mo, the multiplicities of maximal singularities, and the number of maxirnal singularities) then the “untwisting” process finally terminates in an isomorphism. This classical schcrrre works perfectly in the case of surfaces. One can get a detailed notion about it frorn Iskovskikh (1991b), where Crernona t,ransforrnations of a plane PE are “untwisted” over an a,lgebraically non-closed perfect field Ic (although, as was already rnentioned, the method has been known since the middle of the previous century). In dimension 3 and higher, one rneet,s serious difficulties in trying to apply this scheme. First, because there exist flops and flips (these birational maps are isomorphisms in codimension 1) Second, elementary “uritwisting” maps sometimes lead to singular variet,ies, not to mention that finding and constructing elementary “mltwisting” maps become much more difficult. This is the reason why one can hardly find in the classical literature serious complete results on this subject, alt,hough there are rich experimental data.

59.1. Birational

Automorphisms

of Fano Varieties

173

It is Mori theory that gives an adequate and conceptually clear formalism for untwisting birational maps (see Corti (1995), Ried (1991), Sarkisov (1989) and Sect. 9.2 below) when elementary untwisting maps are constructed using extremal (in the sense of Mori) contractions, flips, flops and their inverses. But, this universal approach has its own additional difficulties. First,, one should work in the category of varieties with terminal singularities, where multiplieities of base cornponent,s, generally speaking, are not integers (thus, even the question of the termination of the untwisting process becomes nontrivial, see Theorem 9.2.8 below). Second, the definition of elementary birat,ional transformations, being conceptually transparent but not constructive, leaves aside difficult, questions of constructing them in concrete situations. In dirnension 2, this general approach gives nothing but a new point of view. But in dimension 3 and higher, it is undoubtedly useful as a general rnethod for finding untwisting rnaps in concrete situations, for example, for Cremona transformations of p’. The investigat,ions in the 2-dimensional case are practically completed (see Marlin (1972)) Iskovskikh (1979c), IskovskikhTregub (1991)) Iskovskikh-Kabdykairov-Tregub (1993) and also the survey of Manin-Tsfasman (1986) for further references), and it is they that suggested the right, setting of the problem within the framework of Mori theory (Iskovskikh (1996b)). The problem was stated in such a setting for the first time by Sarkisov (1989), where the finiteness theorem for the process of factorization in dimension three was also announced. The precise formalization was given later by Ried (1991); the proof of the Sarkisov fact,orization theorern using the latest advances in three-dimensional Mori theory was given in Corti (1995). Examples 9.1.4. (i) Let X, X’ he three-dimensional Fano varieties with p(X) = p(X’) = 1. Let 7’, r.’ be their indices, and let H, H’ be positive generators of the groups Pit(X) and Pic(X’) respectively. Let x: X - - + X’ be a birational map, and let cp*H’

= o*nH

- c

u,c~&.~E,,,

n,

v, t 25:

n

> 1

in the notation of (9.1.1). Here rnn = $, and, by Lcmrna 9.1.1 with m = f, we get that for n > 5 there exists a maximal singularity Bi-1 of multiplicity u, such that,: a) if B,-1 is a point, then vi > F-; in this case the point, a,-l,o(Bi-l) = B E X is also a base point of multiplicity v(B) > v, > $, because multiplicities do not increase under appropriate resolution; b) if 13,-1 is a point, then v, > F: and the following t,wo cases should be dist,inguished: bl) ai-l,o(&l) = B is also a curve; then v(B) > I/, > F; b2) (infinitely near case) ai-l.cj(B,pl) = B E X is a point of multiplicity u(B) > u; > $. There

exist, t,he following

obvious

restrictions

for maximal

multiplicities:

174 0)

Chapter

Y. Rationality

Questions

for Fano Varieties

II

Let Ci , , C,s be curves on X which are images of all the maximal curves (maxirnal singularities of X), and let deg C, = C;. H. Then C;‘=, deg C, < r.‘H”. This follows immediately from 0 ( ((P*H’)~ . o*H > 0 by the projection formula:

which implies C deg C, < r2 H”. p) Let Brj E X he a point of the greatest, multiplicity vi (it is not necessarily a maximal singularit,y in the sense of our definition): then ~1 < n@. This follows again from (p*H’)” (o*H - ‘T~,~&) > 0:

Hence

~1 ( YL@.

Actually,

t,his inequality

is usually

st,rict.

In particular, if X = V4 c p4 is a quartic, then C deg C; 5 3, and vi < 2ra, that is, there do not exist, maximal points (and, as was shown in IskovskikhManin (1971), other maximal singularities cannot exist eit,her). If X = p’, then deg C; < 16: and vi < ‘n in case n) For nonsingular maximal curves of Cremona transformations of p’; the elementary untwisting maps were found in the classical literature. This results were reconstructed by nl. Gizat,ullin and S. Tregub (unpublished). For maximal curves of birational transformations of a three-dimensional cubic, elementary rnaps were listed in Tregub (1984a). (ii) Let 7r: X 4 5’ be a standard conic bundle over a rational surface 5’ wit,11 Pit(X) = 7r* Pit(S) + Z. (-Kx), and let C c S, C # 8 be the degeneration curve (the discriminant curve). Let X: X- - +X’ be a birational map orno a smooth three-dimensional variety X’, and let H’ be a. numerica.lly effect,ive divisor on X’ such that, IH’+mKx,l = (D f or all m > 1. In the notation of (9.1.1), let

where D E Pit(S), and 0,: b E Z. b 2 0. By Lemma 9.1.1, if b 2 1, and thert are no maximal singularities (that is, there are neither curves of multiplicity 7/i > b nor points of multiplicity v, > 2b), then the linear systern I~*01 is empty, and t,hcrefore IDI is also empty. If maximal singularities exist, then, as was shown in Sarkisov (1980), one can get rid of them (or at any rate get rid of some of them) by blowing up points on the base with a subsequent, base extension and elrmenta~ry hirational transformations along tonics over the base. Therefore one can assume that the linear system IDI is empty. As

59.1. Birational

Automorphisms

of Fano Varieties

175

was shown in Sarkisov (1980), this fact contradicts the assumption b > 1 if 141(~ + Cl # @ (that is, the only possible cases are when either b = 0 or X is an isomorphism on the general fiber of 7r). From this it follows that X is nonrationa.1, and has neither other conic burldle structures (in the birational sense) nor birational maps onto Fano variet,ies or de1 Pezzo fibrations. This result ca.n be used in pa,rticular to construct examples of non-rational t,hree-dimensional varieties with trivial intermediate Jacobian (see Sarkisov (1980)). The twodimensional prototype of this result over a non-closed field was obtained in Iskovskikh (1967). (iii) Now let S: X - P’ be a de1 Pczzo fibration of dimension 3 (thai is, the general fiber of b is a de1 Pezzo surface) of degree d = Kgt, where Ft = b-l(t) is is the fiber over the general point t E lY‘. Assume also that this fibration standard in a sense tha,t X is nonsingular, 0x(-l(x) is relat,ively ample, and Pit(X) = 6* Pic(P’) + Z. (-Kx). Let X: X- - +X’ be a birational map onto a nonsingular three-dimensional variety X’. Let, H’ be a numerically effective divisor on X’ such that lH’ + rnK,y, 1 = 0 for any m > 1. In the notation of (9.1.1): let p*H’

= ag*Ft

- ba’Kx

- xv,o*E;,

b > 0.

By Lemma 9.1.1, if b > 1; and if H = g*(p* H’ has no maximal singularities (that is, either curves of multiplicity v, > b or points of multiplicity v, > 26), then a < 0. This means that if the linear system \H’I is movable, then the linear system I - bKxl should also be sufficiently movable. This is rather a strong restriction and can hold only in exceptional cases. Based on t,his consideration, the problem of rationality of de1 Pezzo fibrations with twodimensional fibers of degree d = 1,2,3 is discussed in Iskovskikh (1995).* Remark 9.1.5. There exists the following numerical version of Lemma 9.1.1. Assurne that for some m > rn(i the divisor H’+mKxf is essentially numerically non-effective, that is, for some algebraic family of curves covering the whole of X’, the intersection of H’ + mKx, with the general curve of the farnily, say 2, is negative. Then for every such rn > m. either there exists a maximal singularity or the divisor H + rnKx is not numerically effective. The proof (in a classical marmer) is also very simple. Namely, in the notation of (9.1.1), we have:

a*(H

+ mKx).

p*Z

+ (CjrrrS;

- I/,)&E,)

t y*Z

Now if vi 5 rn&, then (C(mfi, - v 7) o&,,,E,i) . cp*Z > 0, and cr*(H + rnKx) . cp*Z < 0. Therefore H + mKx is not numerically effective, because a,cp*Z is also a general curve of a family of curves covering the whole of X. In such a * Pukhlikov (1998) recently tions of general position.

solved this problem

for d = 1,2,3 under some assump-

Chapter

176

9. Rationality

Questions

for

Fano

Varieties

II

form the NoetherPFano lemma can be generalized to any Q-Fan0 fibrations 7r: X + S with rational nb(j: m, 6,. V, (see Corti (1995) and Lemma 9.2.6 below). It is the numerical non-effectiveness of divisor D that is used in Sarkisov (1980) to prove the main result described in Example 9.1.5 (ii). Now we formulate the principal results on birational maps of Fano varieties which were obtained by means of t,he classical NoetherPFano untwisting method. We do not present here the corresponding results for rat,ional surfaces over perfect fields (see the survey of Manin-Tsfasrnan (1986) and also Iskovskikh (1991b), I 5,k ovskikh (1979c), Iskovskikh-Tregub (1991), IskovskikhKabdykairov-Tregub (1993)) Iskovskikh (199610)).

Theorem (i)

(ii)

Let X be a nonsingular three-dimensional Fano var%etiy of index r = 1 urd w h’h’ac as 0 f one of the follouiin,g types: p(X)=1 (1) a quark in p”; (2) a double cover of p” ram,ijied along a sextic; an,d let x: X- - -X’ be a hirational map onto a nonkngular variety X’ uhich, is either a Fano variety, or a stundard conic bundle, or u stan,dard de1 Pezzo ,fibrution (that is, X’ is a nonsingular three-dimensional Fano fibration). Then X’ is also of the same type as X, and x is an isomorphism. In particular, X is n,ot rational, and Bir(X) = Aut(X) (see Iskovskikh-Manin (1971), Iskovskikh (1979b)). Let X be a three-dimensional Fan,0 variety of index 2 and degree d = 1 (that is, a double cover of the Veronese cone), an,d let x: X - - +X’ bc u hirational map as in the previous case. Then either x is an isomorph,ism, or X’ is a Fano variety of index 1 and p = 2, an,d x-l = 0~: X’ + X is a blow-up of some fiber of the rutionul map I+-+ Kx, : X - - + p2. In particular X is not rational, and Bir(X) = Aut(X) (see Iskovskikh (1 YYYb),

(iii)

9.1.6.

Khashin

(1984)).

Let X be a Fano uurietr/ ,which l;s a, double cover of a nons%n,gular threedirnensionul quad+ Q (see Table 12.2, No. 2), and let x: X- ~ +X’ be a birataonal map as above. Then th,ere exists u birutional a,wtonborph~sm $: X - - +X such that x Q $1 X + X’ is un isomorphism. For every line Z c X (that is, -Kx.Z = 1, a,ndp,(Z) = 0; such, curves constst,ute a onedimensional fumily), th,ere is an ussociated involution TZTZ: X ~ ~ +X with Z being a maxim,al singular%ty pif Z does not lie on the ramification divisor). Then $ is a composition, of such3 involution,s, an,d the group Bir(X) cun be represented us un extension: 1 --i B(X) where

B(X)

by the

involutions

cation

---) Bir(X)

--7‘ Aut(X)

+

I ,

is the free product of the 9roups { 1, Y-Z} of order 2 generated TZ for all lines Z C X which do not lie on the rum+ I n p ar t ZCIL _ 1ar, X is not rational. divisor (see Iskouskikh~ (19796)).

39.1. Birational (iv

of E’ano Varieties

177

Let X be a Fano variety which, is an intersection of a quad% FL and a cubic Fs in P”, and let x: X - - -,X1 be a birutional map as above. Then there exists a hirational automorphism II/: X - - +X such, that x o $: X + X’ is an komorphism. For every line Z c X or, respectively, for every conic C c X th,e plane p(C) of which is con,tuined in Fx (the families of lines and such conks are on,e-dimensional), there are associated involution,s QZ and PC, respectively. If C decomposes into two lines or is u double line, then & can be expressed in terms of OZ, which gives some relations can be represented as un extefns%on between QZ. Th,en the group Bir(X) l+

(v)

Automorphisrns

B(X)

+ Bir(X)

---f Aut(X)

+

1,

where B(X) is the quotient of the free product of th,e groups of order modulo some explicitly described relations. In two {l,az} and {l,Pc} particular, X is not rational (see Iskovskikh (19796), Pukhlikov (1989)). Let X be an n-dimensional Funo variety of index r = 1 wh,ich is one of the following:

(1) n quintic (2) a double (3) a double und let x:X-

(vi)

in p”; cover ofp”? n 2 3; cooe’r of a qundric Q c iY’+l, n > 4; - +X’ be a birationul map as above. Then x l;s an Gomorphism. In particular, X is not rational, and Bir(X) = Ad(X) (see Pui%hlikov (1987), Pukhliko,u (19883)). Let X C p4 be a quark with a single ordinary double point x0 E X such that all the 24 lines kying on X and passin,g through x0 are different. Let x: X ~ +X’ be a birationul map as above. Then if X + X is the blow-up of x(~, and X: X - - + X’ is the in,duced m,ap, then there exists u birutional nutom,orphism

The ysro,up Bir(X)

$1 X - - + X can

such

th,at

be represented

1 + B(X)

--f &r(X)

x o3 : X us un

+

x’

is an

isomorphism.

extension:

---f Ad(X)

+ 1 ,

where B(X) = n&{l,rL} is the free product of 24 groups each generated by the involu,tion deterlnined by one of the 24 lines and the group generated by the involution, 725 determined by the double cover X 4 p’ (the projection from x~~). In purticular, X is not rational (see Pukhlikov (1988a)). Remarks 9.1.7. (i) All t,he preceding results remain t,rur if one takes for X’ any Q-Fano f&ration (see, for example, Corti (1995) for the case of a t,hree-dirnensioria.l yuartic). (ii) The maximal singularities and the a.ssociated elemedary untwisting maps for a three-dimensional Fano variet,y of index 2 and p = 1 were found in Khashin (1985): and for a three-dimensiond cubic in Tregub (1984a). Unlike the cases described above, for a cubic some elementary untwisting maps are not hirat,ional automorphisms. For example, there are elementary birational

Chapter

178

9. Rationality

Questions

for Fano Varieties

II

maps onto a Fano variety VI4 c p” connected with a normal rational curve of degree 4 or with a quintic of genus 1 (see Tregub (1985), Takeuchi (1989): for these Iskovskikh (1979b), Logachev (1983) and Chap. 4). Unfort,unately, types of Fano varieties, the question of existence of infirmely near maximal singularities is not yet solved.

$9.2. Decomposition in the Context

of Birational Maps of Mori Theory

9.2.2. Here we describe briefly the approach of Sarkisov (1989) t,o the study of birational maps between @Farm fibrations from the point of view of Tori theory. We follow Ried (1991) and Corti (1995). We shall work essentially in dimension 3. R.ecall that a Q-Fan0 fibrution is an irreducible projective variety X with at most Q-factorial terminal singularities together with a cont,raction in the sense of h4ori theory, that is, with a morphism ‘p: X -+ S onto a variety S with diniS < dirnX, p(X/S) = 1, and -Kx relatively p-ample. In particular, in dirnension 3 it is either generically a conic bundle over a surface S, or a de1 Pezzo fibration over a smooth curve, or a @Fan0 variety with /, = 1 and a structure morphism y: X --f pt (the contraction of the whole of X into a point). As in the log theory, we shall consider pairs (X, hH), where X is a. variety, H is a linear system without fixed components, and b E Q is a non-negative rational number. We recall the following Definition (i)

9.2.2

(cf Sect.

1.1).

A pair larities and

(X, bH) h as. t erminal (respectively if for every birational morphism f-exceptional divisors E,, the equality KY + bf$H

h,olds

with

transform fixed

(ii)

= f* ( Kx

a% > 0 (respectively of the

components),

linear and

ui

system

canonical, f: Y

X

+ bH) + c

> 0: a~ >

H (that

log

4

is,

-1). the

canonical)

with

singu-

nonsin,gular

Y

aiEi

Here f;‘H inverse

totul

is the

proper

image

minus

CL, E Q.

Assume that X has at most terminal singularities. The can,onical threshold of the pair (X, H) is the maximal positive rationul number c = c(X, H) such that the pair (X, cH) h,as canonical singularities. Similarly,

ber N such

the

log canonicul

that the pair

The following fact is known Kollbr et al. (1992)).

threshold

(X, OH) from

is th,e maximal h,as log

minimal

canonical

rnodel

positive

,ratiorbal

nurn-

skgularities.

t,heory

(see, for example,

$9.2. Decomposition Proposition Let H be a linear Mori-extremul pair

(Z,

cHz)

of Birational

HZ

of Mori

Theory

179

9.2.3.

Let X hawe at most Q-factorial termind singularities. without fixed components on X. Then there exists a divisor-id contraction p: Z + X and a number c such that the has canonical singularities, an,d system

Kz+cHz where

Maps in the Context

is the

proper

=p*(Kx+cH),

tr,n,sform

on, Z

of th,e &near

system

H.

It is clear t,hat c’ is the canonical threshold for (X, H). We shall morphism p: Z + X the muximul blowup. Consider now the category of three-dimensional Q-Fan0 fibrations S with morphisms which are arbitrary birational maps X:

call the cp: X +

X--X-*X/

S

S’

not necessa,rily coordinated with the morphisms ply call such X maps or links (as in Ried (1991), Hom(X/S, Xl/S’) 1s ’ 1 no t empty, then we shall say The general problem is to decompose any link into mentary links. By analogy with the two-dimensional t,o define an elementary link to be q o p -r from the

cp and cp’. We shall simCorti (1995)). If the set that X and X’ are linked. a composition of some elecase, it would be natural following diagram:

;-.Z< X

P1 S

X’

I

p’

S’

where p, q are ext,remal divisorial contractjions (perhaps q or both of them may be isomorphisms) as in the case cont,ra,st to the two-dimensional case, in dimension three flops, and it turns out that they should be included in the

trivial, that is, p or of surfaces. But in there exist flips and following definition.

Definition 9.2.4 (see Corti (1995)). A morphism in the category of threedimensional Q-Fan0 fibrations deJned above is called elementary (lir~k) if it belongs to on,e of the following Pypes. Type I. Th,is is the commutative diagram:

180

Chapter

where

Z

+

X

a composition, S’

+

9. Rationality

1:s an

S is a usual

Type II.

This

divisorin

extremal

Mori

of

Questions

flips

surjective is the

and

morphism.

Fano

Z

+

Z - - + Z’

is

Type III.

where an

Xextremal

morphism.

Type IV.

where

p(S/T)

II

contraction,

(that

is,

inverses

diagram

+ S where

Varieties

and Z- - +X’ is to Mori flips). Here we have p(S’/S) = 1. Tori

anti-flips

commutative

for

+ S

X and 2’ + X’ are extrem,al divisorial Mori con,tractions, a composition of Mori flips and anti-flips, and S’ r” S. This is the commutative diagram

~ + Z

is

a composition

Mori

divisorial

of

Mori

contraction,

jlfps and anti-flips, Z + X’ and S + S’ is a usual surjective

is

Here we have p(S’/S) = 1. This is the commutative diagram

X - - +X’

= p(S’/T)

is a composition

of Mori

flips

and anti-flips.

Here

we have

fibration

on X,

= 1.

Links of type IV always change the structure as the following lemma shows (Corti (1995)).

of a Q-Fan0

Lemma 9.2.5. Let X/S and X’/S’ he th,ree-dimensional Q-Fan0 fibraLet x: X - - +X’ be a birational map wh,ich is an isomorphism in codimension 1 (for example, a comjposition of flips, flops an,d anti-flips). Assume that there exists a rational map $1: S- - + S’ such that the following diagram tions.

PI 9’ I s--“‘-+I

39.2.

Decomposition

Then

is commutative.

of Birational

Maps

in the

Context

of Mori

Theory

181

li/: S - - -) S’ is hiregulur.

To be able to urltwist any birat’ional map between Q-Fano fibrations, one needs: of course. an ana,logue of the Noether-Fano lemma. for this sit,uatjion. Let

X--X-+X’ P I s

I

10’

S’

be any birational map. Choose a very ample divisor A’ on S’ and a sufficiently large positive integer p’ such that the linear systerrr H’ = 1 - $Kx, + ,‘*A’1 is very ample on X’. Denot’e by H = x,~,l(H’) the proper transform on X of the linear system H’. Then for sorne positjive rational number p and some (not necessarily ample) divisor class A on 5’. we have: H=-pKx+cp*A. Let p: (2, HZ)

+ (X, H) 1)c a resolution.

(9.2.1) Then

we have the diagram: HZ = q*H’

X H F /LK~

+ cp*A

X’

9

1 S’

I S

P’

H’ = 1 - /LLK~~ + p’*A’I

Since here IL is not necessarily an integer, but only a positive rational number, therefore the simple decreasing of IL’S, while untwisting by means of elementary links, could be insufficient for the terminatiorr of the process. But in dimension 3 the numbers p have bounded denominators, namely: if X ---i S is a conic bundle, t,hen p E $Z; if X + S is a de1 Pezzo fibration, then p E $iz U ;Z; if X is a Q-Fan0 variety with p = 1, then the boundedness of denominators of /L’S follows from the boundedness of indices (see Kawamata (1992a)). Definition. In the preceding notation, let c denote the canon,ical of th,e pair (X, H). We shall say that H bus a, maximal singularity For the following Lemma 9.2.6 precedin,g notation: (9 (ii)

lemma,

see Ried

(NoetherpFano

(1991), inequality

Corti

threshold if c < i.

(1995)

in the Q-Farm

category).

171 th,e

map P 2 P’, and th,r: equality holds if and only if x induces a rational s- - +S’; if Kx + i H has canonical singularities und is nwnekx@y effective, then x is an isomorph~ism, ad it induces an isomorphism S cx S’; in particular, /A==‘.

182

Chapter

9. Rationality

Questions

for Fano Varieties

II

This lemma is proved in a way similar to that, of the numerical analogue of the classical Noet,her-Fano lemma described in Remark 9.1.5 with the use of the standard techniques frorn rninirnal model theory. To apply this lernrna to t,he process of decomposition of birational maps into a composition of elemcnt,ary links, one should establish parameters of a map which should decrease. Definition. iso~‘s the

In

dyree) canonical

threshold

blow-ups

crepant pliciby

b).

birational

the

precedin,g

is defined

or,

morph,ism

the

triple

(pL; c, e),

of the pair

(that what

notation,

to be th,e

(X,

H),

degree

and

such

that

e = p(Z/X),

Kz

+ cHz

1-1 is as in

e is the

is, th,e nmm,ber of maxCna,l

is equivalent,

(H, qx X + S) (Sark-

of

rwhere

= p* ( Kx

c is

of

singularities where

(9.2.1),

(Kx + cH)of H of multi-

number p: Z ---) X

is a, maximal

+ cH).

These triples are lexicographically ordered, that is, (pl: cl, el) < (p, c, e) if ~1 < p, or LL~ = p and cl > c, or /-LI = p, cl = c and el < e. The problem is t,o construct an elementary link

such that the degree (/LI, cl, el) of the birational 5’1) is less than (cl, c, e). Theorem (i)

9.2.7

(Corti

for (Iii,

pl, XI

+

(1995)).

If H has a maximal singularity, II gives (~1, cl, el) < (p, c, e). in a commutative diagram

where

rnap x0x;’

TTT:5’1 + S is a birational of general fibers.

then

untwisting

Moreover,

morphism,

with

~1 < p

if

and

links ~1

x1

can,not

~1,):

X,

+

of type I or be included

X1?,

is

an

isomorphism

(ii)

If H has no maximal singularity, then some sequen,ce of types III and IV either decreases p or term,inates

of elementary at an

links

isomorphism.

By virtue of this theorem, to prove t’he existence of a decomposition of a hirational map into a sequence of elementary links it is sufficient t,o prove the finiteness of the decomposition process. The lat,ter is clear for elementary links of t,ypes III and IV, because links of type IV are compositions of flips, and for flips we have the finiteness theorem. Links of type III decrease the Picard number . For the remaining cases there is the following

slO.1.

Some

Constructions

of Unirationality

183

7% we is no in&ite Theorem 9.2.8 (Corti (1995)). ,seyuence of links &: xk/&. - - + &.+,/s k+l of type I and II each beginning with u maximal blow-up pk: zk 4 Xk with canonical Kz, + c~Hz,, = p*(Kxk + q.HxI. ), Ck < j$ = ;. In other words. according to Thr>orem 9.2.8, the sequence ck has no accum~~l&ion points from below. This fact is derived from the following result from rninirnal rnodel theory (Alexeev (1993), Koll& et al. (1992)). Proposition 9.2.9. Let A be the set of all log can,onical thresholds (Y = H), where X is a three-dim,en,sional variety with a,t most termin,al singularities, and H is a linear system without fixed compon,ents. Then in A c Iw th,ere au-e no accumsulation points from, below. (Y(X,

As a result,

we get the proof

of the following

Theorem 9.2.10. Every birational Fano jibrations can be decom,posed i&o

staternent

map between a composition

of V. Sarkisov. three-dimension,al of elementary

Qlinks.

Remark 9.2.11. In the two-dimensional case7 say over any perfect field, the analogous result is more or less obvious, because here the rnaxirnal multiplicities f are non-negative integers. In a series of papers on the birational theory of rationa,l surfaces (see Iskovskikh (1991b), Iskovskikh-KabdykairovTregub (1993), Iskovskikh-Tregub (1991) and further references in the survey of Manin-Tsfasman (1986)): more deta,iled resultIs were obtained. Namely, for every particular type of surfaces, explicit constructions of elementary links are given, and defining relations between them are also described. These results were summarized in Iskovskikh (1996b).

Chapter 10 Some General Constructions and Unirationality 510.1. Some Constructions

of Rationality

of Unirationality

The rnost frequently used approach is the following: let f: X ~ + Y be a rational dominant map of irreducible varieties. Suppose tha,t there exists a dominant morphisrn p: Z + Y such that the fibcred product X xz Y i XxzY.X

184

Chapter

10. General

has a unirational onto X. Then X, following Enriques

Constructions

of Rationality

and Unirationality

component W c X xz Y which is mapped dominantly Y, 2 are mlirational. An illustration of the above is the criterion for the unirationality of conic bundles.

Proposition 10.1.1. Let f:X + Y he a conic bundle over u unirational base Y. Then X is unirutional if and only if there exists a unirutional suboariety Z c X such that the morphism f 1~: Z + Y is domin,ant. Indeed, X xz Y + 2 is a conic bundle over 2 having some rational section. Therefore X x zY is hirationally equivalent to 2 x @ and is unirational since Z is unirational, whence X is unirational. Conversely, if X is unirational, then there exists a dominant rational map $: l?dixnX - ~ +X finite in the general point. Further, the irnage of a sufficiently general (dirnX - l)-dimensional subspacc E c IPdimX in X is the desired subvariety Z c X. Since unirational surfaces arc rational (over an algebraically closed field of characteristic zero), then in the case dimX = 3 the surfaces Y and X from Proposition 10.1.1 should be rational. Examples 10.1.2. (i) Let X c pTa+‘, n > 2, be a cubic hypersurface, a,nd let 1 c X be a line on X. After the blow-up 0: X’ + X of 1, the projection q:x--*p-l from 1 turns X into a. conic bundle X’ + p7L-1. The exceptional divisor E = up1 (1) is a 2-section for t,his conic bundle. Since E is a rational variety, then X is urlirational with the degree of rational cover being equal to 2. (ii) Let X C p” be a complete intersection of three quadrics. Denote the web of quadrics passing through X by n r” p2. Choose a line 1 c X and denote by p/ ‘v P4 the space of two-dirnensional linear subspa.ces in P” passing through the line 1. Let Ql(X) = {(q,p) E I7 x PI 1 p c q}. The projection K: &l(X) + 17 = p2 is a conic bundle with the discrirninant, curve C c p2 of degree 7. Denote by Xl the blow-up of X along the line 1. Let Y = {(q, x) e 17 x Xl I (1:x) c 4}, where (1, x) is the plane spanned by the line 1 and the point 2. Consider the following dia.grarn

where cw(q,x) = 2, and /3(4,x) = (q, (1,x)). The morphisms (1 and p are birational; therefore X is birationally equivalent to the conic bundle &i(X). The exceptional divisor E = a-‘(l) of the blow-up (T: Xl + X of the line 1 is rnapped surjectively with degree 4 onto 17 y p”, whence X is unirational with the degree of the rational cover being equal to 4. To make the picture complete, we reproduce here the classical construction (set, for example, Roth (1949)) of the unirationality of X c p” coinciding in fact with the previous one. The projection from a sufficiently general line 1 c X maps X birationally onto a quartic X’ c p4 which contains a ruled surface of degree three R y F1,

5 10.1.

Some Constructions

of Unirationality

185

the image of the blown-up line 1. The two-dimensional family of planes in P4 generated by the two-dimensional linear system of tonics lying on R gives on X’ another two-dimensional family of residual tonics. This turns X’ into a conic bundle (with isolated singularities) over P2. Fibers of this conic bundle intersect R in four points. Thus, the surface R is a $-section, which implies the unirationality of X with degree 4. (iii) Let X = XJ c IP4 be a quartic with a line of double singularities 1 c X. Let 0: Xl ----f X be a resolution of singularities along 1. Then 7r: Xl + lP* is a conic bundle, and the surface E = 0~~~ (1), the proper transform of I, is rational and is a 4-sect,ion of 7r Therefore X is unirational (see Come-Murre (1977)). To prove the unirationality of some uniruled and rationally connected rieties X, the following considerations are usually applied. Let P s-x

va-

be a covering family (that is, the rnorphism p is dominant) of ra,tional curves on X. Assume that there exists a unirational subvariet,y Y 2 T such that some irreducible component 2 c q-l(T) has a rational section over Y, and plz: Z- ~ +X is dominant. Then X is obviously unirational. In connection with this, we would like t,o mention one interesting classical construction of unirationality connected with farnilies of elliptic curves of degree n covering the whole of X (see Roth (1955)). A ssume that the subfarnily of such curves passing through a general point x E X is one-dimensional and is parametrized by a rational curve. For a general elliptic curve E passing through x, there exists a unique point ‘y such that the divisor (n - 1)x + y on E is a hyperplane section. Therefore for a sufficiently general point z E X, there exists a rational curve C(x) on X swept out by points y while E varies in the one-dimensional rational farnily of elliptic curves passing through IC. Assume tha.t this procedure applied to points of a fixed curve C(x) gives some surface F(z) which is obviously rational. If for points z E F(z), the curves C(Z) cover a dense subset in X, then X is unirational. Examples i0.1.3. (i) Let X = Vi’ c P” be a complete intersection of a quadric Q and a cubic F:s. Denote by M the base of one of the two irreducible families of two-dimensional planes on Q. Let P + 111 be the corresponding family. It is known that M is a smooth rational variety of dimension 3, and P is a locally trivial bundle over M. Let 5 E X be a sufficiently general point. Then through X passes a pencil of planes from M parametrized by P1. These planes cut out on X a one-dimensional farnily of cubic curves passing through Z. Let E be a curve from this family. It lies in the corresponding plane IP from the family M. The tangent line to E at the point z intersects E in one other Restrict it to X. point z’. The natural projection 7r: P + Q is a @-bundle

186

Chapt,er

10.

General

Constructions

of R.ationality

and

Unirationality

Then P = T-‘(X) is a ~l-bur~dle over X, and the preceding construction gives a rational dominant map P- - -X. To prove the unirationality of X: it suffices to take any rationa. surface Y c X such that, 2 = n-‘(Y) is mapped dominantly onto X (see Roth (1949), Chap. IV). (ii) Let X he a three-dimensional Fano variety which is a double cover of a qua.dric Q c p’. Let p: X + Q be t,he corresponding cover with t,he ramification divisor D = QnVb being thtl int,ersection of Q and a quartic V, c p4. Through every point z E X passes a one-dimensional family of elliptic curves which are inverse images of lines on Q. This family is pa.rametrized by a conic C(x) which is the base of the tangent cone for Q at the point cp(x) E Q. Fix some integer m 2 2. Let H t / - h’x 1 be an a,nticanonica.l divisor. Then for every ellipt)ic curve E from t)hc family C(x), the linear system lmH El on E cont,ains the unique effective divisor of the form (2m - 1)~ + y: where ?/ E E c X is some point. When E runs over the family C(:r), the points ~1 sweep out a rat,ional curve on X isomorphic to C(.x). Now; as in the> general classical construction described above, one ca.n establish the mlira.tionality of X (see Iskovskikh (1979,)). (iii) Let, X be a double cover of p3 ramified along a. smooth quartic D c p”. Inverse images of lines form a four-dimensional family of elliptic curves on X. For a sufficiently general “line” 1 c X, the subfamily of these cllipt,ic curves intersecting 1 has dimension 3, and through a genera.1 point J: E X passes a one-dimensional family of such curves. Therefore X is unirational by the same reason as above (see Iskovskikh (1979b)). We remark that the unirationality of X can also be obtained in the following way. The inverse image of the pencil of planes passing through a sufficiently general line in p3 is a pencil of de1 Pezzo surfaces of degree 2. The general fiber of the pencil is a de1 Pezzo surface of degree 2 over a non-closed field k(pl) and has k(I?l)-point,s. Therefore it is unirational over k(n’) (see Manin-Tsfasman (1986)); whence X is unirational. (iv) (The construction of B. S’e~~re (see, for example; Iskovskikh-Marlin (1971)). Let, X C p4 1)e a smooth quartic. For a sufficiently general z E X, t,hc base of the tangent, cone to the tangent hyperpla.ne section H(z) at the point 5 E X is a smoot,h conic. Then for an open subset X0 c X, we have a conic bundle 7r: So + X0. Define a rational map p: 5’0 - - +X which sends a point, s E So to the point y E X which is t,he fourth intersection point of X a.nd the ruling of the tangent cone t,o H(z) at the point 2 corresponding to s. Every such ruling of the tangent cone to H(z) is eit,her a triple tangent to X at z or lies in X. Since for almost all points :1: t X, only a finit,e number of triple tangent lines at 2 lie in X, the map p is well-defined for almost all points s E So. The basic idea of B. Segre is to construct a three-dimensional rat,ional subvariety Zo c So such t’hat the restriction of p to Z, is dominant. It turns out that for some quartics t,his can be done, namely, it is possible to find a rational surface Fo c X0 such that, Zo = ~-‘(Fcj) possesses the desired property.

§lO.l.

Some Constructions

of Unirationality

187

To do t,his, it suffices to choose X and a rational surface F c X in such a way that the conic bundle r/z”: Z’n + Fe, where Fo = F n XoP has a section: and the map p: 2” - - +X is dominant. In the examples constructed by Segre, one takes for F a hyperplane section with a triple point ~0 E F c X. Therefore F is rat,ional. For a general point z E F, the t,angent plane to F a.t II: intersects the quadric tangent cone H(z) along two ruling (or one double ruling). These two rulings give two points in the corresponding fiber of the conic bundle 7r: 20 + Fo. In other words, there is a 2-section of this conic bundle. One should choose X and F in such a way that this 2-section splits. For this, it is sufficient that the discriminant of the equation defining two triple tangent lines to a general point ~1: E F be a square as a function depending on z t F. An explicit example of such X and F is given in Segre (1960) (see also Iskovskikh-Manin (1971)) Iskovskikh (197913)): x

:

xi + x0x;

F

:

I$ - Gzfz;

+ x;I - 6~~~;

+ x; + xi + x&,

= 0;

+ x; + x; + IC;Q = 0 ,

that is, F is a hyperpla,ne section of X determined by the equation ~0 = 0. The surface F has the unique triple point (0, O,O, 1). The discriminant of the equation for two triple tangent lines coincides up to a square factor with the t,hereforc the surface Hessian of t,he form F. It is equal to (36( ~9 + x$).c~)‘; of triple tangents to X decomposes over F. It, is unknown whether a, general quartic in p4 is unirational. The constructions considered above can be used to obtain result for tllree-dimerlsional Fano varieties. Theorem cept perhaps (i) (ii) (iii) (iv)

10.1.4. All three-dimensional for the follow@ ones:

Funo varieties

the following

are unirationul

ex-

X = VJ, u double cover of P” ramijied along a seztic (Table 12.2, No. 1); X = VI, u, double Veronese cone (Table 12.2, No. 11); X = Vd, a general quark in P4: (Tuble 12.2, No. 2); VI, the blow-up alon,g an elliptk cu,rzle (““zi) - 1, that is, when a general variety of this type contains a linear subspacc lY”‘-’ (the projection from this subspace gives a birational isomorphism X - - + p”), and is unirational if 71 > $ - 2, t,hat is, when it contains a linear subspace p”‘-’ of unirationality is a (see Roth (1955)). W e remark that this construction particular case of the general MorinPredonzan construction of unirationality of complete intersections, which is considered in the next section.

188

Chapter

10. General

310.2.

Constructions

Unirationality

of Rationality

of Complete

and Unirationality

Intersections

10.2.1. The classical result of Morin (1940) states that hypersurfaces of a fixed degree d become unirational starting from a sufficiently large dimension. The reason of this is that they contain linear subspaces of sufficiently large dimension. Predonzan (1949) extended these statements to complet,e inters&ions. These results of U. Morin and A. Predonzan were reconstructed in a series of contemporary papers (see Ramanujam (1972), Ciliberto (1980), Kollar et al. (1992)). Roughly speaking, the basic idea is very simple: counting of dimensions shows that for any positive integer T, a complete intersection X c IIr’” of fixed multidegree and sufficiently large dimension contains a linear L cut out on X the family subspace L = p’ Subspaces l?r+’ C p” coinaining of subvarieties Y in such a way that X n pr+’ = L + Y. The variet,y Y is also a complete intersection of srnaller degree, and the whole family is unirational by the induction hypothesis. As we have alrea.dy seen, a cubic hypersurfa.ce X = X3 c p’“, 71 > 3, gives the simplest example. In this case one can take for L a line L C X Planes p2 passing through L cut out on X a family of tonics paramet,rized by IED’“-2. This conic burldle is unirational because it has a rational 2-section which is the exceptional divisor of the blow-up of the line L. Following Kollar et al. (1992)) we illustrate the basic idea of the proof with the example of a quartic X = X4 c p”. First we calculate when a general (and therefore every) smooth quartic contains a linear subspace L P P3. Let pN be the space of all quartics in pTL. Let G = Gr(3, n) be the Grassmannian of three-dirnensional subspaces in P”, and let 2 c G x pN be the subvariety of pairs (L, X), L E p’, L c X. We have the following diagram: Z-GxPN

l/I

G

IPN.

The subvariety Z has codimension (4zs = 35, and the variety G has dimension 4( n> - 3). Therefore dim 2 > dim p II if 4(n - 3) > 35. One can show (see Kollar et, al. (1992)) that under this condition the map Z + pN is surjective, that is, every quartic X contains some linear subspace L F IP”. Next, for a given pair L c X, four-dimensional subspaces p” c pn containing L cut out on X, in addition to L, an (n - 2)-dimensional family of cubits Y c p4 with intersections Y n L = P” c X which are cubic surfaces. Let 0: XL + X be the blow-up of X along L, and let E be the exceptional divisor. We have a natural map ‘p: XI, + prL-‘, whose fibers are cubic hypersurfaces Y. The fibers of 91~ = $J: E + pTLp4 are cubic surfaces. If for the general point 7 E pne4, the cubic hypersurface Y, = cp-i(r) is unirational over the field k(r) = k(prLp4), th en X is obviously unirational over Ic. To prove the unirationality of Y, over k(r), it is sufficient to prove that there

310.2. Unirationality

of Complete

Intersections

189

exists a line on Y, which is defined over some field K > k(r) rational over I; (that is, K is isomorphic to the field k(V) for some rational variety V with will be satisfied if the a dominant map V + IPne4). I n t urn, these conditions general cubic surface F, = 4)-l(r) contains a line defined over K. It, remains now to find such a field h’. Let Gr( 1,3) be the Grassrnannian of lines in p’, and let V c Gr( 1,3) x P”-’ be the subvariety {(l,t) 1 1 E Gr(l,3), t E PP4, 1 c p-‘(t)} of lines 1 contained in fibers of the morphism $: E + PP4. The morphism V ---) PP4 is surjective since every cubic surface contains lines. Now if the map V + Gr(l,3) is sur.jective, t,hen it is a P-bundle for some s because the fiber over a point 1 E Gr(l,3) is the linear projective space of cubic- surfaces from the family d!: E ---f PnP4 pa.ssing through 1 in P”. Therefore V is rational since the Grassmannian is a rational variety. To get, the rnap V + Gr(l, 3) to be surjectivc, it is sufficient to take n big enough such that, say, the family $I: E i IPP4 includes all cubic surfaces in p’, that is, the ma.p PP4 + P’” (@a is the complete family of cubits in P”) should be surjective. If X is sufficiently general, then it is easy to show (see Kollar et al. (1992)) that the inequality n > 23 guarantees the fulfillment, of this condition. (this is a particular case of inequality (10.2.2) below). Finally, it is clear t,hat F, = $-‘(r): and therefore Y, = (p-l(r) contains a line over K = k(V) > k(r). This proves the unirationality of a quartic in IV” for n > 23. We turn now to the general case. Let d = (dl, . , cl,,,), dl 5 d2 < . . . < d,,, be a set of positive integers, and let n, r be positive integers such that one of the following conditions holds: (i) if nr = 1 and di = 2, then T-L > 2r; (ii) if m, > 1 and d # (1, 1, . , 1,2), then (n - r)(r

the following

+ 1) > 2 i=l

(“I”>

inequality .

holds: (10.2.1)

Proposition 10.2.2 (Kollar et al. (1992)). Let n, sr, dl , , d,,, be positive integers as above, and let one of the conditions (i), (ii) above be satisfied. Let Hi be hypersurfuces of degree d, in P” . Then there exists a linear ssubspace L ry P’ c IP”” which is contained in the intersection X = HI n . n HT1, of all the Hi. Introduce

now the following

notation.

Definition 10.2.3. For positive integers n, rJ dl, . . . , d,,, dI < da < . < d,,,, we define the integers r(d) = r(dl,. ,d,,,) and n(d) = n(dl,. . ,d,,,) by induction in the follomin,g way: (i) (ii)

if if

r = 1 and dl = 1, then n,(l) = 1 and r(1) = 0; r > 1 and dl = 1, then n(l, d2, . . . , d,,,) = r(L dz, . . ,d,,,) = r(&, . ,d,,):

1 + n(&,

. , d,)

and

190 (iii)

Chapter

10. General

Constructions

of Rationality

$fd, > 1 f or all %,we set d - 1 = (dl - 1,.

r(d)

(‘(“i,“;-

1x1

that

. : d,, - 1) und

de,fine

= n(d - 1) ;

n,(d) = r(d) + 2 We remark

and Unirationality

‘)

7

for all n > n(d) and T = r(d)

we have the inequality (10.2.2)

Now we are ready Predonzan (1949)).

Theorem

to formulate

the Morin-Predonzan

result

(Morin

(1940),

10.2.4 (KollBr

(X, L) be a general of hypersurfuces H, space of dimension ulong L. Then the

et al. (1992)). In, the preceding notation, let X = HI n.. .flH,,, is the complete intersection3 in IV’ of degree d; respectively, und L is a linear projective r contuined in X. Let X he irreducible und noasingulur variety X is unirutionml for n, > n,(d) and r = r(d).

puir,

where

The pair (X, L) is genera,1 in the following sense. Let 2 = {(HI,. . . , H,,,L) E Int(rtl,dl:. ,d,,,) x Gr(r,n) 1 L c HI n.. n H,,,}, where Int(n, dl, . : d,,,) is the space of all m-tuples of hypersurfaces in p’” of degrees dl, , d,,, respectively. Let z E Z be a point. Denote by X, the intersection of the rn hypersurfaccs H;, . . . , H,“,, corresponding to z and by L, c X, the corresponding projective subspace. Let iY(n, d) be the open subset in Z consisting of points z for which the following conditions hold: (i) X, is irreducible (ii) the different,ial the point z.

and nonsingular of the map Z -

along L,; Int,(n, dl,

. , d,,,) has maximal

Then for any point z E U(n,d) Theorern 10.2.4 is true. More exact bounds for n and r were obtained in Ramero mulate them, we need t,he following additional notation: e(dl , holds; r’(1) = r’(d) = T’(d) =

. , d,,, , r) is the minimal

integer

n for which

r(l) = 0, n’(l) = n(l) = 1, r’(2) = 0, n,‘(2) r(d), n’(d) = n(d) if dl = 1; and max{n’(d - 1) - 1, e(d - 1, r’(d ~ l))},

I,& n’(d) = 1 + C [(“‘(~~7~~-‘)

(1990).

the inequality

rank

in

To for(10.2.1)

= 2,

- I] if dl > 1.

I= 1 Finally we set N(d) = max{e(d, r’(d)), t,rur for n3 > N(d) (see R.arnero (1990)).

n’(d)}.

Then

Theorem

10.2.4

is

rjlO.3.

Some

General

Constructions

of Rationality

191

Remarks 10.2.5. (i) For the case of a quartic X c IV, we have r(4) = 61, ‘n(4) = 23725, so the bound for n3 in the Theorem 10.2.4 is rather rough. In 10.2.1 we obtained the bound n > 23. The refinerncnt of Theorem 10.2.4 proved in (Ramero (1990)) g’ives r’(4) = 3, n’(4) = 20, e(4,3) = 12, and N(4) = 20. It is known (see, for example, Roth (1955), Cilibcrto (1980)) that U. Morin proved t,he unirationality of a genera.1 quartic for n3 2 7, t,hat is, when it contains P”, and the unirationality of a general quintic for n 2 17, that is. when it contains p’. Some general conditions of unirationality of hypersurfaccs are also discussed in (hlorin (1954)). (ii) It is clear that all unirational and rational complete int,ersections (in particular, hypersurfaces) are Fano varieties.

510.3.

Some

General

Constructions

of’ R,ationality

The simplest rational varieties after projective spaces are irreducible projection frorn any nonsingular point quadrics Q c l??IL+i. the stereographic J: E Q gives a birational isomorphism Q ~ - + lY”. Immediately from the defintions, one can establish the rationality of Grassmannians and, more generally, flag varieties, many homogeneous and quasi-homogeneous spaces with respect to actions of linear algebraic groups. For example, all toric varieties arc rational. The most frequently used classical construction of rationality of projective varieties is the projection frorn linear subspaces. Exarnple~ 10.3.1. (i) Let X c l? 7L+‘r’ be a genera.1 complete intersection of m quadrics. If ‘II > (“‘zl) - 1, t,hen X is rational (see Roth (1955)). Indeed, if n 2 (“‘zi) - 1, then X cornains a linear subspace L = IPpl (see 10.2.2). For a general subspacc E”“’ c lYfrn passing through L , the intersection lFnxLU{. r } consists of L and a point z which is t,he intersection point of m projective subspa.ces lYrJpl in p”“. These subspaces are cut out on X by the quadrics defining X. Therefore the projection KL: X - - -p” is a hirational map. (ii) The simplest examples of singular rational varieties are hypersurfaces X c lY’+l of degree d which have a (d ~ 1)-fold point. The projection from this point is a birational isornorphisrn X- ~ +J?‘. There are many examples of singular rational Fa.no varictics (hypersurfaces, cornplctc intersections, etc.) which contain linear subspaces of sufficiently large dimension, in particular, a plane in the three-dimensiona. case (see Rlzat,i-Bertolini (1988). Alzat,i-Bertolini (1989), Alzati-Bcrtolini (1992)) Roth (1955)). For exa.mple, the question of rationality is completely studied in Alzati-Bcrtolini (1988), Alzati-Bertolini (1989), Ambrogio-Rornagnoli (1989) for cubic complexes V, in p’ (that is, complete intersections of a quadric and a cubic) which contain p > 1 planes p2 rnccting each ot,her exactly at a point, (such a, Vfi is singular because, by the Lefschetz theorem, for a nonsingular V, the Picard group is

Chapter

192

10. General

Constructions

generated by the class of a hyperplane for p > 4 and non-rational for p < 3.

of Rationality section

and Unirationality

-Kv,),

Such Ve are rational

To prove the rationality in these and many ot,her examples, the following general method is applied: the initial variety is represented birationally as a fibration over a rational base in such a way that the generic fiber is rational over the function field of the base, for example, as a locally trivial p’-bundle or as a conic bundle with a rational section. For example, VG containing two disjoint planes is rational, because the projection from one of the pla.nes makes from V, a conic bundle over @ with the second plane being a rational section. In connect,ion with this, the following result providing sufficient (and presurnably necessary) conditions for rationality of conic bundles in dirnension 3 proved to be useful. Proposition 10.3.2 rr: V + S be a stnn,durd

(see Iskovskikh conic

bundle

(1991a), over

(1987)).

surface

S

Let with,

con-

if th,e dl;vl;sor Ks + C is not numerically effecttiue, except for the case when TT: V + S is birutionully equivalent (us a conic bundle) to the cork bundle ~0: q/;, 4 IP2 obtained from a nonsingular three-dimensionul cubic via the projection from a line. nected

discriminunt

CUIW

C

c

S.

Then

Iskovskikh

a rational

V is ro,tional

This result is used in Alzati-Bertolini (1992) to prove the rationality of some three-dirnensional Fano varieties with p > 2 and in Alzati-Bertolini (1989) to prove the rationality of cubic complexes VG c p” with p > 4 planes. In Kolpakov-Prokhorov (1992), Kolpakov-Prokhorov (1991) it is used to prove the rationalit,y of some singular three-dimensional Fano varieties of type p’/G where G is a finite group acting linearly on p”. The following st)atement, is proved in Kolpakov-Prokhorov (1992), Kolpakov-Prokhorov (1991). Proposition a finite

subgroup,

10.3.3. is rational

The quotient except

ua,ri,ety perhups

for

P”/G,

where

the

following

G c

PGLd(k)

cases:

is AG, the natural G E

G E AT, G E PSLz(F7), G ‘v Ss, G 2 SC, G being the image under map SLz(k) x SLz(k) + PGLd(k) of the product of the binury octuhedrul und binury icosuhedrul groups, and G being an extension of A:, or AG by u group of order 32. In particular, P”/G is rutionul if G is a finite solvuble group.

We rernark that if G c PGL n+l (k) is an abelian group, then p’“/G is a toric Fano variety with Q-factorial log terminal singularities (see 2.1.10), which is obviously rationa,l. As we have seen in Chap. 4, projections frorn points, lines, tonics, double projections from points and lines, and some other elementary rational maps can be used to prove the rationality of many types of three-dimensional Fano varieties with p = 1. Combining these results with the results on the rationality of three-dirnensional Fano varieties with p > 2 (see Alzati-Bert,olini (1992)), we obtain the following general fact. Theorem 10.3.4. All three-dimensional Funo varieties are rational except the following ones: No. l-5, 7, 1 l-13 f rom Tuble IS.2 (Cha,p. lg), No. 1 3, 5, 6, 8, 11 from Table 12.3 (Chup. 12), No. 1 from, Table 12.4 (Chap. 12).

for

510.3.

Some General

Constructions

of Rationality

193

To prove the rationality of some three-dimensional Fa.no varieties X, the following standard scheme due to F. Enriques is frequently used: first one tries to find (birationa.lly) on X a pencil of rational surfaces, and then to prove the rationality of the general member of the pencil. The latter can be done using the following criterion of rationality of rational surfaces over an algebraically non-closed perfect field (see Iskovskikh (1979c), Manin-Tsfasman (1986)) Iskovskikh (1996b)). Proposition 10.3.5. Let F he a m,inim,al r,tion,al ,surface cwer a perfect field k (the word “rational” means in this situation that F = F@% is birational to iPt over the algebraic closure % of k). Then F is rational over k (or, *what is the same, is birutionally trivial over k, see Manin-Tsfasman (1986)) if and only if the following conditions a’re satisjied: (i) F has a k-point; (ii) d = K: > 5. In our situation, Ic is the function field of a curve defined over an algebraically closed field, so condition (i) is always satisfied (Manin-Tsfasman (1986)). Returning t,o hypersurfaces, we remark that t,he general (in the sense of moduli) hypersurface X c PTr+’ of degree d > 3 presuma.bly is not, rational obviously except for the case of cubic surfa.ces (n = 2, d = 3). Nevertheless in the case d = 3, there are some known constructions of rat,ional nonsingular cubits of higher dimensions (see Roth (1955)) Tregub (1984b). Tregub (1993)). Examples 10.3.6. (i) The first const,ruction generalizes the following classical construction of rationality of cubic surfaces. On a cubic surface X c P”, we choose two disjoint lines L1: L2 c X (it is known that such lines exist on every nonsingular cubic surface). The line L c P”, L g X, passing through two points 21 E L1, 22 E L2 intersects X in exactly one other point z e X. For a general point x E p’, there is exactly one line L with this property. As a result, we obtain a birational map IP’ x P1 - - -X which establishes the rationality of X. Now let X c Pn+’ be a cubic hypersurface of even dimension n = 2r > 2. Assume that X contains two disjoint subspaces L1 = P’, L2 Y P“. .Just as in the case of surfaces, we have a birationa,l map IP’ x P- - +X. By the Deligne-Noether theory, the Chow component A” @ Q of cycles of middle dimension r > 1 on a general cubic X, dimX = 2r, is generat,ed by the class of a linear section. Therefore the general cubic X cannot contain two disjoint linear subspaces P” (see Zarhin (1990)). Alternatively, one can see that the general cubic X c lP”“+l does not contain P”, r 2 2, immediately by counting dimensions (see 10.2.2). But for every r > 2, there are nonsingular cubits X, dirn X = n = 2r, which contain two disjoint subspaces P’, for example, Fermat cubits:

194

Chapter

10.

General

Constructions

of Rationality

and

Unirationality

rr+1 x

Indeed, if C is a cube root linear equations det,ermine

:

c(-l)%p

= 0.

of unity, < # 1, then the following two systems two disjoint linear subspaces iY’ c X:

zo - 21 = 22 - x3 = . . . = 2,, - z&,+1 50 -

Cx*

= 22 -

ransl.: fvInt,h. USSR, Izv. 20 (1983) 23552571. Zhl. 566.14014

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