Springer algebraic geometry 5 parshin shafarevich (eds) springer

Mori on extremal properties of cones of effective one-dimensional cycles Mori 1982, Mori 1988, using which the concept of a minimal model playing the central role in the classical birat

Editor-in-Chief

R V Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute,

ul Gubkina 8, 117966 Moscow; Institute for Scientific Information (VINITD,

ul Usievicha 20a, 125219 Moscow, Russia, e-mail: gam @ipsun.ras.ru

V A Iskovskikh, Steklov Mathematical Institute, ul Gubkina 8,

117966 Moscow, e-mail: iskovsk @mi.ras.ru

Author and Translator

Yu G Prokhorov, Moscow State University, Department of Mathematics,

Algebra Section, Vorobievy Gory, 117234 Moscow, Russia,

e-mail: prokhoro@mech.msu.su

Translator

S Tregub, 70115 Tashkent, Uzbekistan

Fano Varieties

V A Iskovskikh and Yu G Prokhorov

Translated from the Russian

by Yu G Prokhorov and 5 Tregub

Contents

Chapter 1 Preliminaries 0.00.0 0c cece ee ee eee 7

§1.2 On Numerical Geometry of Cycles cu 11

§1.3 On the Mori Minimal Model Prograi 13

$1.4 Results on Minimal Models in Dimension Three 17

Chapter 2 Basic Properties of Fano Varieties .-.- 23

82.1 Definitions, Examples and the Simplest Properties 23

82.2 Some General Ñesulis cece ee 34 82.3 Existence of Good Divisors in the Fundamental Linear System 39

82.4 Base Points in the Fundamental Linear System 47

Chapter 3 Del Pezzo Varieties and Fano Varieties of Large Index 50

83.1 On Some Preliminary Results of Fujita .00-5 50 83.2 Del Pezzo Varieties Definition and Preliminary Results 53

§3.3 Nonsingular del Pezzo Varieties Statement of the Main Theorem and Beginning of the ProoŸ co 54 §3.4 Del Pezzo Varieties with Picard Number p = 1 Continuation of the Proof of the Main Theorem 57

§3.5 Del Pezzo Varieties with Picard Number p > 2 Conclusion of the Proof of the Main Theorem 62

Chapter 4 Fano Threefolds with p= 1 0 0 cee es 65

84.1 Elementary Rational Maps: Preliminary Results 65

54.2 EFamilies of Lines and Conics on Fano Threefolds 71

84.3 Elementary Rational Maps with Center along a Line 76

$4.4 Elementary Rational Maps with Center along a Conic 36

94.5 Elementary Rational Maps with Center at a Point 95

94,6 Some Other Rational Maps c2 so 101 Chapter 5 Fano Varieties of Coindex 3 with p = 1: The Vector Bundle Method Quy 104 85.1 Fano Threefolds of Genus 6 and 8: Gushels Approach 104

85.2 Á Review of Mukais Results on the Classification of Fano Manifolds of Coindex 3 0.0 000000000 cece eee ee 108 Chapter 6 Boundedness and Rational Connectedness D0000 i05 Ta tenes 116 86.1 Uniruledness nrn aaaaá da .- ences 116 §6.2 Ratlonal Connectedness of Fano Varletles 120

Chapter 7 Fano Varieties with Ø0 2Ð 2 ng và 128 §7.1 Fano Threefolds with Picard Number p > 2 (Survey of Results Of Mori and Mukal tees 128 87.2 A Survey of Results about Higher-dimensional Fano Varieties with Picard Number Ø0 3 2 Q2 nh ao 141 Chapter 8 Rationality Questions for Eano Varietles Ï 153

88.1 Intermediate Jaecobian and Prym Varietfios 153

88.2 Intermediate Jacobian: the Abel-lacobi Map 162

§8.3 The Brauer Group as a Birational Invariant 166

Chapter 9 Rationality Questions for Fano Varicties I] 170

89.1 Birational Automorphisins of Eano Varlietles 170

89.2 Decomposition of Birational Maps in the Context of Mori Theory 0.0.0 eee 178 Chapter 10 Some General Constructions of Rationality and Unirationality 00.0 cee ete eee 183 §10.1 Some Construections of Unirationallty cu cuc cà 183 910.2 Unirationality of Complet© HifGTSGGEONS cu cu 188 §10.3 Some General ConstrucHons of Rationality 191

Contents 3

Chapter 11 Some Particular Results and Open Problems 196

§11.1 On the Classification of Three-dimensional Q-Fano Varieties 196 611.2 GeneralizatlONS ch nh nh nh nh h nh nh es 203 611.3 Some Particular Ñesults che hen hen nh nhớ 208

§11.4 Some Open Proble1ns - ch nh hen nh nh nhở 212 Chapter 12 Appendix: Tables -.- ch nh hen 214 812.1 Del Pezzo Mamifolds ch nh nh nhe 214

§12.2 Fano Threefolds with ø= Ì ch nh nh nh nhớ 214

§12.3 Fano Threefolds with p= 2.0.0.0 0 nh nh nh nh nh nhớ 217

§12.4 Fano Threefolds with p= 3 0555.0 cee eee eee eee 220

§12.5 Fano Threefolds with ø= 4 che hen nh th sở 223

§12.6 Fano Threefolds with ø@ > 5 {ch nhe nh nh hen 224 812.7 Fano Fourfolds of Index 2 with p 3 2 ch nh nh nh 225

§12.8 Toric Fano Threefolds - {nh nh nh hnn 226

Introduction

This survey continues the series of surveys devoted to the classification of al- gebraic varieties (Shokurov (1988), Shokurov (1989), Danilov (1988) Danilov

(1989), Iskovskikh-Shafarevich (1989), Kulikov-Kurchanov (1989)) It deals_

with Fano varieties of dimension three and higher The general classification 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 Mori theory of minimal models It is based on the re-

markable ideas and results due to S Mori on extremal properties of cones of

effective one-dimensional cycles (Mori (1982), Mori (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 dimension Within the framework of the theory arises the category of projective varieties with some admissible singularities: terminal canonical, log canonical and others

A minimal model in the sense of Mori is defined to be a normal projec- tive variety with a numerically effective canonical divisor According to the

Mori Minimal Model Program, which is completely carried out in dimen-

sions < 3 and partly in dimensions > 4 (see Mori (1982), Reid (1983a), Kawamata-Matsuda-Matsuki (1987), Clemens-Kollar-Mori (1988), Kollar et

al (1992), Wilson (1987a)), every irreducible algebraic variety over an alge-

braically closed field of characteristic zero is birationally equivalent either to a

minimal model (if its Kodaira dimension > 0) or to a fibration over a variety

of smaller 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 —oc)

Therefore the Mori program establishes the important role that Fano vari- eties 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 —oo

The only one-dimensional Fano varieties are the projective line over an algebraically closed field and a conic over an arbitrary field Two-dimensional Fano varieties are del Pezzo surfaces (see the survey of Iskovskikh-Shafarevich (1989))

In connection with the problems of rationality and unirationality, G Fano studied at the beginning of the century the class of varieties with canoni-

cal curve-sections (see Fano (1908), Fano (1915), Fano (1930), Fano (1931), Fano (1936), Fano (1942), Fano (1947)) Contemporary authors continued

this study, taking for the definition of the class of varieties the ampleness of the anticanonical sheaf G Fano did not restrict himself to considering only nonsingular varieties, but for the present only nonsingular three-dimensional Fano varieties are classified in contemporary works Although the problems

of rationality and unirationality still remain important (and very difficult),

OU

Introduction

at present it is the general problems of the structure theory that are of prime interest, namely, the classification of Q-Fano varieties with admissi- ble singularities in dimension three and higher, the problem of bounded- ness of the degree, solved for nonsingular Fano varieties of any dimension

in Kollar-Miyaoka-Mori (1992c), Nadel (1991), Campana (1991a), the prob-

lems of uniruledness and rational connectedness, also solved for nonsingular

Fano varieties of any dimension (see Miyaoka-Mori (1986), Kollár-Miyaoka-

Mori (1992c), Campana (1992)), the study of Fano varieties with additional

structures (IP’-bundles, toric varieties, and others; see Batyrev (1981), Demin

(1980), Szurek-Wisniewski (1990c), Wisniewski (1989b), Wisniewski (1993))

There are several rather complete expositions of the classification theory for nonsingular three-dimensional Fano varieties (see, for example, Iskovskikh

(1979a), Iskovskikh (1988), Murre (1982), Mori-Mukai (1986), Mori-Mukai (1983a), Mukai (1992a)) Singular Fano 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 sep- arate results and to highlight the main directions and methods of research

We do not include in this survey the well-known (actually classical) results on two-dimensional Fano varieties, that is, nonsingular del Pezzo surfaces (see, for example, Nagata (1960), Manin (1972)), and on del Pezzo surfaces with canon-

ical (see Du Val (1934), Demazure (1980), Hidaka-Watanabe (1981), Brenton

(1980)) and log terminal singularities (see Alexeev (1988), Alexeev-Nikulin (1989), Nikulin (1989a), Nikulin (1988), Nikulin (1989), Alexeev (1994b)} We

do not touch arithmetic results for Fano varicties (see Manin-Tsfasman (1986), Batyrev-Manin (1990), Manin (1993)), and the few known results in charac- teristic p > 0 (see Ballico (1989), Serpico (1980), Shepherd-Barron (1997))

The ground field & is assumed to be algebraically closed and of characteristic Zero

The Russian version of this survey was finished in 1995 and unfortunately many 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 Chapter 1

In Chapter 2 we give the basic definitions and examples, and formulate

the simplest properties of Fano varieties which can be immediately deduced from the definition and general theorems such as the Riemann -Roch theo- rem, vanishing theorems etc We also include in this chapter some general results on equations defining varieties connected with Fano varieties (canoni- cal 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 systems and on their base locus

Chapter 3 is devoted to the description of the results due to T Fujita on the classification of polarized del Pezzo varieties connected with n-dimensional Fano varieties of index n — 1 We give also the proof of the main classification theorem

In Chapter 4 we present the classification of three-dimensional Fano vari- eties with Picard number p = 1 The classification is based on the general- ization of the classical method due to G Fano of a double projection from a line, a conic, etc with the 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 g = 1 using vector bundles The method

of vector bundles makes it possible to 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 connectedness and the bounded-

ness 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 8—

10 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 Artin In Chapter 9 we consider the method

of factorization of birational maps (the classical Noether-Fano 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 del Pezzo varieties and nonsingular three- dimensional 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 Beltrametti for their hospitality and the opportunity to work in excellent conditions He also thanks the Itai- ian Consiglio Nazionale delle Ricerche (CNR) for the financial support The second author thanks the fund “Pro Mathematica” for the financial support The present work was also partly financed by Grant No M30000 from the International Science Foundation and by the Russian foundation for funda- mental researches (project 93-011-1539)

The authors would like to thank N A Valueva for her help in preparation the manuscript

§1.1 Singularities 7

Chapter 1

Preliminaries

$1.1 Singularities

Let X be an irreducible normal projective variety of dimension dim X = d

over an algebraically closed field & of characteristic zero In this survey we shall

use the following notation (see Kawamata-Matsuda-Matsuki (1987), Clemens- Kollár-Mori (1988))

Z,(X) is the group of r-dimensional cycles on X, that is, the free abelian group generated by closed irreducible subvarieties of dimension r,0 < r < d—1 In particular, Z;_,(X) is the group of Weil divisors on X

Div(X) is the group of Cartier divisors on X There is a natural injection

(since X is normal)

Div(X) — Za-1(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 Zy_1(X)®Q are called Q-divisors, and elements of the group Div(X) & Q are called Q- Cartier divisors

Pic(X) denotes, as usual, the Picard group, that is, the group of classes

of Cartier divisors with respect to linear equivalence This group is naturally isomorphic to the group of invertible sheaves (or, equivalently, line bundles)

on X up to isomorphism

A QCartier divisor D € Div(X) ® Q is said to be big if h°(X,Ox(mD)) >

const -m® for (sufficiently large) m >> 0 such that mD € Div(X)

A Q-Cartier divisor D € Div(X) & Q is called nef (numerically effective) if D-C > 0 for every complete curve C Cc X The intersection number is defined

as a rational number in the following way: let m € Z be an integer such that

Cycles z, 2’ € Z|(X) are said to be numerically equivalent, which is written

as z= z’,if L-z= 1-2’ for every L € Pic(X) By duality, one can define numerical equivalence in Pic(X) The pairing Pic(X) x Z;(X) — Z induces a

perfect pairing

where N!(X) := Pic(X)/(mod =) @R, and Ni (X) := 2 (X)/(mod =) @ R

For Pic(X)/(mod =) & Q, we use the notation NG(X)

Let w be a rational differential form of degree d on X Then one can define

the Cartier divisor of this form (w)|y 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 canonical and

is denoted by Ky or simply by Kk

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 € Pic(X), that is, Kx is represented by a Cartier divisor (the converse in general is not true, see Ishii (1987), Ishii (1991)) If mA’x € Pic(X) for some integer m, then X is called Q-Gorenstein The minimal positive integer m with this property is called the (Gorenstein) index of Kx (or X) All smooth

varieties are, of course, Gorenstein

Example 1.1.1 Let Fy C P° be the Veronese surface, that is, the image of P? in P® under the map determined by the complete linear system of conics Op2(2), and let X C P® be a cone over Fy Then Pic(X) = Z-H, where H isa hyperplane section Denote by E Cc X the Weil divisor which is the cone over the image of a line 1 € P? in Fy Then —Kx = 3E ¢ Pic(X), so the singularity

in the vertex is not Gorenstein But 2Ky = —6£ = —2H € Pic(X), which

means that X is Q-Gorenstein of index 2 Note that locally near the vertex, X

can be represented as C3/r, where r is the involution (x, y, z) > (—-2, —y, —z)

The expression (dx A dy A dz)*” = (da? A dy? A dz3)”®” /64a2u2z2 defines a 2- canonical form on X If f: X > X is the blow-up of the vertex P € X, and

if F = f-\(P) ~ P® is the exceptional divisor, then X is nonsingular, and

Kx = f*Kx + $F

Definition 1.1.2 A normal variety X is said to have at most canoni- cal (respectively, terminal, log terminal, log canonical) singularities if it is Q-Gorenstein, and for every resolution of singularities f: X' — X with excep- tional divisors E; CX", the following conditions hold:

mK x: = f*(mKx)+ So aj Bi (1.1.2)

>>], +: > |

(m is the index of X) with a; > 0 (respectively, a; > 0

Usually this formula is divided by m and is written as

Kx: = ft(Kx) +o aki, a;i= —€Q, aj

rh

where a; > 0 for canonical (respectively, a; > 0 for terminal, a; > —1 for log

terminal, and a; > —1 fer log canonical) singularities The numbers a; are called discrepancies at E;; they depend only on X and the (proper images of) divisors E;, that is, they do not depend on the choice of resolution

Let wx = O(Kx) be the canonical (dualizing) sheaf For any positive integer 7, we denote by wil the double dual sheaf of the sheaf wor (it is taken

to kill torsion and co-torsion, see Reid (1980b)) Then wt is a torsion-free

§1.1 Singularities 9

sheaf of rank 1: it is locally free if and only if i = am for some integer a > 0, where m is the index of Ky If X has at most canonical singularities, and f:X!' — X is a resolution, then f.(w%!) = wil for i > 0

Proposition 1.1.3 (see, for example, Clemens-Kolldr-Mori (1988)) (i) In dimension two, terminal points are nonsingular

(ii) Two-dimensional canonical singularities are exactly the Du Val ones (they are also called rational double points) Locally in the complex topology they can be determined by one of the following equations:

There is a complete list of terminal singularities in dimension 3 (refer to Mori (1985), Ried (1987), Kollar (1991)):

Theorem 1.1.4

(i) Three-dimensional terminal singularities are isolated points

(ii) A three-dimensional hypersurface (i e Gorenstein) singularity is terminal

if and only if it is isolated and is defined by an equation of the form:

g(z,,Z) + th(œ,,z,t) =0,

where g is one of the equations (1.1.8) Such singularities are usually

called (compound Du Val) cDV-points (see Reid (1980b), Ried (1987))

(iii) Every three-dimensional terminal singularity is a quotient of some hy- persurface terminal singularity (which is called a canonical (m: 1)-cover, where m is the index of Kx at the singular point) by some cyclic group The typical situation is:

(cyt fe", =0)C C!/Z„(1, —1,a,0), (am) =1,

where CN /Zm(a1,- ,an) denotes the quotient of C% by the cyclic group action (%4, 0N) > (C'21, ,C¢% an), and ¢ is a primitive root of unity

of degree m Exceptional cases can be written in the form:

(2? + fly, 2,t) = 0) Cc C*/Zm (a,b, ¢, d) for some m <4 There is a complete list of all possible cases

(iv) Every three-dimensional terminal singularity can be deformed to a set

of terminal cyclic quotient singularities of the form C?/Zn(1,—1,a), (a,m) = 1

Remarks 1.1.5 (i) Two-dimensional log terminal singularities were stud-

ied in Brieskorn (1968), Iliev (1986), Kolldr et al (1992) They are exactly

quotient singularities of (C?,0) by finite group actions For log canonical sin- gularities, see Kawamata (1988), Kolldr et al (1992)

(Gi) The singularity in the vertex of a cone over the Veronese surface which was considered in Example 1.1.1 is terminal of index 2: it is isomorphic to the quotient singularity C?/Z4(1, 1,1) (see, for example, Wilson (1987a))

(iii) All log terminal (in particular, terminal or canonical) singularities of any dimension are rational, that is for some (every) resolution f: X’ — X, the equalities R' f,Ox = 0,7 > 0, are true (see Elkik (1981), Kawamata-Matsuda- Matsuki (1987)) Log canonical singularities are not necessarily rational even

in dimension 2 (see Kollar et al (1992))

(iv) Terminal singularities form the least possible class of singularities in-

volving which the Mori Minimal Model Program is stated and can be true (as in dimension 3) Canonical singularities are exactly those which arise on canonical models

(v) In the general Minimal Model Program (Kawamata-Matsuda-Matsuki (1987), Kollar et al (1992)), more general types of singularities are defined and used Namely, let D = $° a;D; be a Q-divisor on a normal variety X (D; are irreducible Weil divisors) such that Ay + D is a Q-Cartier divisor Then for every resolution f:.X' — X we have:

In terms of discrepancies of only exceptional divisors, formula (1.1.4) can

be rewritten in the form

Kx + (f7').(D) = f*(Kx +D)+ So aiki, a €Q,

where the star in the subscript denotes the proper image of D as a Weil divisor

A resolution f: X’ — X is called a log resolution of the log variety (X, D) if

§1.2 On Numerical Geometry of Cycles 11

irreducible components of the support Supp((f~1).(D)) are nonsingular and

intersect transversally Refer to Shokurov (1992), Kollar et al (1992) and

Kollar (1997) for different variants of the definition of log terminal objects in this more general situation

§1.2 On Numerical Geometry of Cycles

Let X be a normal projective variety of dimension d A cycle z = 0 nz €

Z4(X) is called effective ifn; > 0 Vt Recall that we denote the numerical

equivalence of 1-cycles with respect to intersections with Cartier divisors (and,

by duality, the numerical equivalence of Q-Cartier divisors) by the symbol =

A variety X is called Q-factorial if some integral multiple of every Weil

divisor is a Cartier divisor, that is, if Z;-1(X) ® Q = Div(X) ® Q

The following notation is standard (see Clemens-Kollár-Mori (1988), Kawamata-Matsuda-Matsuki (1987)):

N(X) := Zi(X)/(mod =) 3 R:

NE(X) is the least convex cone in N(X) containing all effective 1-cycles;

NE is the closure of NE(X) in the real topology; this is the so-called Mori cone of X;

NS(X) is the Néron-Severi group of classes of Cartier divisors with respect

to algebraic equivalence:

p(X) := rk(NS(X)) = dime N(X) is the Picard number

Note that a numerically 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 = R4[z] Cc NE(X),

z € Z,(X) @R is called an extremal ray if:

(i) -—Kx +2 > 0, and ¬

(ii) from 2, + 22 € R, 21,22 € NE(X) it follows that 2) € R and z¿ € KR; this

means that the ray R lies on the boundary of cone NE(X)

A rational curve C CX is called an extremal curve if R,[C] is an extremal

ray and 0< —-Ky-C<d+l

The important invariant of an extremal ray is the number p(R) =

inf{—Kx -C | C C X isa rational curve whose numerical class is [C] € R}

This number is called the length of the extremal ray R

An extremal ray R is called numerically effective if C-D > 0 for every effective irreducible Q-Cartier divisor D and a curve C such that 'C] € R For every Q-Cartier divisor D, we set

NEp(X) = {z € NE(X)|D-z>0};

in particular, we shall call NEx(X) the positive part of the Mori cone, and

NE_ g(X) = {1z © NE(X) | Kx -z < 0} the negative part By definition,

all extremal rays lie in the negative part of the Mori cone The following important result was proved by Kleiman (1962)

Theorem 1.2.1 (Kleiman’s criterion for ampleness) A Q-Cartier divisor

D ona variety X is ample if and only if D-z > 0 for every z © NE(X)— {0} For the anticanonical divisor —Kx on a three-dimensional variety X, the ampleness criterion takes on the following simpler form

Theorem 1.2.2 (Matsuki (1987)) Let X be a complete normal variety of dimension 3 with at most canonical singularities Assume that |— mK x| 40

for some integer m > 0 (that is, K(—K x) > 0, where (D) denotes the Kodaira dimension of a divisor D, see the definition below) Then the divisor —Kx is ample if and only if —Kx -C > 0 for every irreducible curve C € Z1(X) The important notion in the Minimal Model Program is the numerical dimen-

sion of a numerically effective Q-Cartier divisor D € Div(X) © Q:

v(D) := max{m | D™ # 0}

If Kx is nef, then v( Kx) is called the numerical dimension of the variety X

and is often denoted by v(X) The numerical dimension is closely related to the Kodaira dimension Recall the definition of the latter

For any Cartier divisor D € Pic(X), denote by yjpj: X +P%™!?!, as

usual, the rational map determined by the complete linear system |D| The Kodaira D-dimension K(X, D) is defined as follows (it is also called the Fitaka

A divisor D € Div(X) is big if K(X, D) =d= dim X

The Kodaira dimension K(X) of X is defined to be equal to K(X’, Kx-),

where X’ is any complete nonsingular variety birationally isomorphic to X

§1.3 On the Mori Minimal Model Program 13

(for example, any desingularization of X) The canonical ring of X is defined

to be R(X) = R(X') = @ A°(X', Ox:(mK x-)) We have

moO K(X) := (transcendence degree of R(X))—1 if R(X) #k;

§1.3 On the Mori Minimal Model Program

It is well-known for nonsingular projective surfaces that a minimal surface can be obtained from any other surface by contracting exceptional curves of the first kind ((—1)-curves) As a result, the following theorem is true (see,

for example, Griffiths-Harris (1978))

Theorem 1.3.1 Every minimal surface X satisfies one of the following

conditions:

(i) Kx is nef, that is, Kx -C > 0 for any curve C CX;

(ii) X is a P!-bundle over a smooth curve I;

(iii) X ~ P?

Theorem 1.3.2 Let X be a smooth projective surface with nef Kx Then U(X) = K(X), and the complete linear system |mKx| is free (that is, it has

no base points and fired components) for n > 0

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), Kawamata-

Matsuda-Matsuki (1987)) The essential difference from dimension 2 is that one should admit singularities on minimal models The most 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 > 4, it still

remains open An analogue of Theorem 1.3.2 for higher dimensions is the abundance conjecture For dimension 3 this conjecture was proved by Miyaoka

(1988), Miyaoka (1988a) and Kawamata (1992c) In this and the next section,

we shall give a brief account of results related to the Minimal Model Program

Definition 1.3.3

(i) Let a € R be a real number The symbol [a] denotes the least integer

>a, and |a| denotes the greatest integer < a, 4 € |a| = [a] is the usual

integral part of a

(ii) Let D= Sl aj, D; be a Q-divisor We set

[D] = [yo aiDi| = Sai] Di ;

P| = [Dadi] = DlalDs

{D} := S (a — |a,|)D; 2s the fractional part of divisor D

For the following theorem, see (Kawamata (1982), Viehweg (1982))

Theorem 1.3.4 (Vanishing Theorem) Let X be a nonsingular proper al-

gebraic variety, and let D = S*a;D; be a nef and big Q-divisor Assume that

the support of the fractional part {D} has only normal crossings Then

H'(X,Øx(Kx+[DỊ)=0, vi>0

If D is an integral ample divisor, this is the well-known Kodaira vanish-

ing theorem This generalization of the Kodaira vanishing theorem is due to

Y Kawamata and E Viehweg It looks rather artificial but is widely used in applications

Theorem 1.3.5 (Non-vanishing Theorem, Shokurov (1985)) Let X be a nonsingular projective variety, and let D be a nef Cartier divisor Let G be a

Q-divisor such that |G] is effective Assume that the Q-divisor aD+G— Kx

is ample for some a € Q, a > 0, and the support of the fractional part {G} has only normal crossings, that is, all its irreducible components are nonsingular and intersect transversally Then

H°(X,Ox(mD + [G])) £0

for any sufficiently large integer m > 0

The divisor G is involved here for some technical reasons important for applications It will not simplify the proof if one takes G = 0

By the preceding vanishing theorem,

H°(X, Ox(mD + [G])) = y(X,Ox(mD + [G)))

For the following theorem, see (Reid (1983b), Shokurov (1985), Kawamata-

Matsuda-Matsuki (1987), Kawamata (1984), Clemens-Kollér-Mori (1988)) Theorem 1.3.6 (Base Point Free Theorem) Let X be a projective vari- ety with at most log terminal singularities Let D be a nef Cartier divisor such that the Q-divisor aD — Kx is nef and big for some a € Q, a > 0 Then the linear system |nD| is free for allm > 0

§1.3 On the Mori Minimal Model Program 15

For the following theorem, see (Kawamata (1984), Kollár (1984),

Kawamata-Matsuda-Matsguki (1987), Clemens-Kollár-Mori (1988))

Theorem 1.3.7 (Rationality Theorem) Let X be a projective variety with at most log terminal singularities Let H be an ample Cartier divisor, and let

Theorem 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 € N\(X)| Kx -z <0};

(ii) NE(X) = NEx(X)+ >> Ri, where {Rj} 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-Kolldr-Mori (1988), Kawamata-

Matsuda-Matsuki (1987))

Theorem 1.3.9 (Contraction Theorem) Let X be a Q-factorial projec- tive variety with at most terminal (respectively, canonical, log terminal) sin- gularities Then the following assertions are true

(i) For every extremal ray RC NE(X), there exists a contraction morphism

†:X — Y such that an irreducible curve C is mapped to a point if and only if [C] € R One can require that f,Ox = Oy, and then f and Y are uniquely determined

(ii) One has only the following possibilities for f and Y:

a) the morphism f is birattonal, and its exceptional locus is a divisor In this case Y is again Q-factorial with terminal (respectively, canonical, log terminal) singularities; such an f ts called a divisorial contraction b) f is birational, and its exceptional locus has codimension > 2 in X In this case Y is not Q-Gorenstein; such an f is called a small extremal

contraction

c) dimY < dim X; the generic fiber F = f—'(n), where 1 € Y is the generic point, has terminal (respectively, canonical, log terminal) sin- gularities and an ample anticanonical class —Kx In such a case f is called a Fano contraction

In all the cases, there exists the following exact sequence:

0 — Pic(Y) — Pic(X) > Z (1.3.1)

The most satisfactory assertion is part a) This is a generalization of con-

tractions of (—1)-curves on surfaces to arbitrary dimensions In this case

the variety Y also satisfies the conditions of the theorem, but rk Pic(Y) =

rk Pic(X) — 1 by formula (1.3.1) Assertion c) reduces the study of X to the

study of varieties F and Y of smaller dimension In particular, it can happen that dim Y = 0, and then X is a so-called Q-Fano variety with terminal (or canonical) singularities Even in the case of dimension 3, the study of such singular Q-Fano varieties is far from being completed

The most difficult part (and new in comparison with the case of dimen- sion 2) is assertion c) To treat this case, new birational operations should be involved, so-called flips, which we shall briefly discuss in the next section Now we come to the basic

Definition 1.3.10 Let X be a Q-factorial variety with at most terminal singularities Then:

(i) X is called a minimal model if Kx is nef;

(ii) if there exists an extremal ray R such that the corresponding contraction f:X — Y decreases the dimension, dimY < dim X, and —K x ts ample

on the generic fiber F, then X is called a relative Fano model

The following conjecture (proved for dimX < 3) is also related to the general Minimal Model Program (see Kawamata-Matsuda-Matsuki (1987)): Conjecture 1.3.11 (abundance conjecture) Let X be a minimal model

Then v(X) = K(X), and the complete linear system |mKx| is free for some

m >> 0 such that mKx is a Cartier divisor

It is easy to prove the following

Proposition 1.3.12 Let X be a minimal model Then:

(i) (Kawamata (1985a)) if v(X) = 0, then |mKx| 4 0 for some m; therefore

there exists an integer n such that nK x ~ 0;

(ii) if v(X) = dim X, then the linear system |mKx| ts free for some m > 0

(this is an immediate consequence of Theorem 1.3.5)

Remarks 1.3.13 (i) In dimension 2 one can assume that X is a nonsingular surface, and extremal rays are only R.[C], where C is one of the following: a) Cis a (—1)-curve;

b) C is a fiber of a P!-bundle X — I;

c) C isa line in P? (see Mori (1982))

Therefore after a number of consecutive contractions of (—1)-curves, one gets either a minimal model in the sense of 1.3.10, or a geometrically ruled surface,

81.4 Results on Minimal Models in Dimension Three 17

class of Q-factorial canonical singularities In fact, many of the theorems from the Minimal Model Program hold for the log terminal category (Kawamata- Matsuda-Matsuki (1987), Kollar et al (1992)) The abundance conjecture is proved only in dimension 3

(iii) Generalizations of Theorem 1.3.2 to the log terminal and log canonical cases in dimension two and three are given in Kollér et al (1992), Keel-

Matsuki-McKernan (1994)

(iv) Kawamata (1985b) showed that the assertion from Conjecture 1.3.11

that the linear system |m4x| is base point free for m > 0 follows from the

equality (X) = K(X), the converse is obvious

(v) From 1.3.8 (Mori (1982)) it follows that if —Kx is ample, then the cone NE(X) = NE(X) is a finitely generated polyhedron every edge of which is

an extremal ray

$1.4 Results on Minimal Models in Dimension Three

The following theorem was proved in Mori (1988)

Theorem 1.4.1 (existence of minimal models in dimension 3) Let X be

a smooth projective three-dimensional algebraic variety over C Then by means

of some sequence of divisorial contractions and flip transformations, the va-

riety X can be birationally mapped onto a projective variety X' belonging to

one of the following classes:

(i) X’ is Q-factorial with at most terminal singularities, and Kx: is nef, that

is, X’ is a minimal model, or

(ii) X’ is a Q-factorial relative Fano model with at most terminal singularities

In case (ti) there exists an extremal contraction f of one of the following

types:

a) f:X' — ¥ is a conic bundle over a normal projective surface Y with at

most rational singularities;

b) f:X' > Y is a del Pezzo fibration (that is, the general fiber F of f is a nonsingular del Pezzo surface) over a smooth curve Y;

c) f:X' — point; here X' is a Q-Fano variety with Pic(X') = Z

Remarks 1.4.2 (i) As was shown in Kawamata-Matsuki (1987), every bira-

tional class of three-dimensional varieties of general type contains only finitely

many minima! models

(ii) In case a) the extremal contraction f: X' -» Y is birationally equivalent (as a bundle) to a “standard form”, that is, to a conic bundle ƒ: X” —¬ Y7

with nonsingular X” and Y’ which is also an extremal contraction (see, for example, Sarkisov (1980), Sarkisov (1982)) (iii) Some particular results on

standard forms for case b) were obtained in Corti (1996)

(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 Q- Fano 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 projective three-dimensional

varieties was obtained by Mori (1982)

Theorem 1.4.3 (Mori (1982), Kollar (1994)) Let X be a smooth three- dimensional projective 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:

(i) R is not numerically effective Then py: X — Y is a divisorial contrac- tion of an irreducible exceptional divisor EF C X onto a curve or a point In addition, yp is the blow-up of the subvariety @(F) (with the reduced structure) All the possible types of extremal rays R which can

occur in this situation are listed in the following table, where H(R) is the length of the extremal ray R (that is, the number min{(—Kx)-C| Ce

R is a rational curve}), and lp is a rational curve such that —Kx «lp =

u(R) and [lp] = R

Type of R | y and BE BCR) | lp

El yp(F) is a smooth curve, and | 1 a fiber of a ruled sur-

Y is a smooth variety face E

E2 y(E) is a point, Y is a | 2 a line on E ~ P?

smooth variety, F ~ P? and

Øg(#) ad Op2{—1)

F3 @(E) is an ordinary double | 1 sxP!orP!xtin E

point, EF ~ P! x P| and Orn(F) > Opr x Pt (-1, —1)

E4 @(E} is a double (cDV)- } 1 a ruling of cone &

point, F is a quadric cone

in P?, and Og(E) ~ Og ®

Opa (—1)

E5 y(E) is a quadruple non- } 1 a line on E ~ P?

Gorenstein point on Y, F ~ P? and Og(E) ~ Op2(—2)

(ii) R is numerically effective Then yp: X — Y is a relative Fano model, Y is nonsingular, dimY < 2, and all the possible situations are the following: a) dimY = 2

§1.4 Results on Minimal Models in Dimension Three 19

C2 vy is a smooth morphism 2 a fiber

b) dimY = 1

DỊ the general fiber of y is a | 1 a line in a fiber

del Pezzo surface of degree d,l<d<6

D2 the general fiber of y is iso- | 2 a line in a fiber

Remarks 1.4.4 (i) In cases F1-E5, Cl, D1-D3, the sequence (1.3.1) is

also exact on the right:

0 — Pic(¥Y) £5 Pie(x) © z — 0

(here (-2) maps a class to its intersection number with /) This is also true in case C2 if surface Y is rational, for example, if h}(Ox) = 0 and |—mKx| #40 for some m € N (Mori-Mukai (1983a))

(ii) In cases £3, 4, F'5 singularities of Y can be written locally analytically

at the point y(£) in the following form:

£3: xy — zt = 0, a Gorenstein cDV-point;

E4: cy — 2° — 8 = 0, a Gorenstein cDV point;

E5: the quotient singularity C3 /Z2(1, 1,1), where Z2(1, 1, 1) is the cyclic group generated by the action (x,y,z) — (—x,—y,—z), this is a terminal point

of index 2, the same as in the vertex of the cone over the Veronese surface

Theorem 1.4.5 (Ando (1985)) Let X be a nonsingular projective vart-

ety Let RC NE(X) be an extremal ray, and let yp: X — Y be its contraction Assume also that py: X > Y has no fibers of dimension > 2 Then:

(i) ify: X =3 Y is a divisorial contraction, then Y is nonsingular, and is

the blow-up of a nonsingular subvariety Z CY of dimension 2;

(ii) ify: X — Y is a Fano contraction, then Y is nonsingular, and y: X + Y

is a standard conic bundle

For the following theorem, see (Ionescu (1986), Wisniewski (1991b), Kawa- mata (1991))

Theorem 1.4.6 Let X be a nonsingular projective variety Let RC

NE(X) be an extremal ray of length p = p(R) Let y:X — Y be tts con-

traction, and let EC X be the set of points where ~ is not an isomorphism Then dim E > $(dimX + —1) Moreover, if p has a non-trivial fiber of

dimension <d, then dim F > dim X +p—d-—1

Corollary 1.4.7 Under the conditions of the preceding theorem, if F is a non-trivial fiber of the morphism yp, then dim F > y—1

Corollary 1.4.8 There do not exist small extremal contractions with

fibers of dimension <1

Now we would like to discuss briefly some new operations in the Minimal 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 “bad” extremal ray and lead to a new Q-factorial variety again with terminal singularities These are the so-called flips

Definition 1.4.9 (Clemens-Kolldr-Mori (1988), Mori (1988)) Let

f:X — Y be a small extremal contraction on a Q-factorial projective variety

X with at most terminal singularities Let E Cc X be the exceptional locus, codim E > 2 A variety X* together with a birational morphism ft:X* > Y

is called a flip if Xt is Q-factorial with at most terminal singularities, f+ is an isomorphism in codimension 1, and Kx+ is relatively ft-ample (note that, by the definition of an extremal ray, —Kx is relatively f-ample, that is, the opera- tion is not symmetric) It 1s clear that the rational map (ƒT) !oƒ: X > XT

in the following diagram

81.4 Results on Minimal Models in Dimension Three 21

Comment 1.4.10 If the flip exists, then either X7 is a minimal model or

it has an extremal ray generated by some numerical class of curves outside

E+, Therefore the contraction process can be continued In the general case

the problem of existence of flips is open In dimension 3 it was solved by Mori

(1988) Earlier Shokurov (1985) proved in dimension 3 that any sequence of

flips terminates The termination in dimension 4 was proved in Kawamata- Matsuda-Matsuki (1987)

V Shokurov (1992) proved the existence of flips in a more general situation:

for three-dimensional log terminal varieties The termination in this case was

proved by Kawamata (1992c) From Shokurov’s proof one can get a new proof

of the Mori theorem (Mori (1988)) Both approaches were thoroughly studied

in (Kollar et al (1992)), in particular, a new proof of the existence of log flips

in dimension 3 was found

Example 1.4.11 (of a three-dimensional flip) This construction was sug-

gested by Clemens-Kollár-Mori (1988), Kollár (1991) Let # c P* be the

image of a geometrically ruled surface F; under the embedding determined

by the linear system |s+2/|, where s is the exceptional section with s? = —1,

and f is the class of a fiber of the projection F; — P! Let Y c P® be the cone over F with vertex at P € Y We have Kp = —2s—3f, and Op(1) = s+2f, so

Kr is not a rational multiple of Op(1), and therefore Oy (Ky) ¢ Pic(Y) © Q Consider the composite of the natural projections Y — P + F, — P!, and let

X+CYxP! cP xP be the closed graph of the resulting rational map

Y —P -+P Then the birational map ft: X + + Y possesses the following

properties:

(i) Xt is a nonsingular three-dimensional variety;

(ii) E+ = (f+)-!(P) ~ P! with the normal sheaf isomorphic to Op:(—1) &

Opi (—2);

(iii) f* | tpt :X+ — E* — Y~P isan isomorphism and (by the adjunction formula) Kx+-E* =1, that is, Ky+ is relatively ft-ample

Next, we blow up curve E+ on X7* and after that the exceptional section

on the exceptional divisor of this blow-up isomorphic to F, The exceptional divisor of the last blow-up is a surface isomorphic to Fy = P! x P! which,

as one can easily see, can be contracted in another 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 exceptional surface F, is isomorphic to P? with normal sheaf Op2(—2) Now this P? can be contracted into a singular point as an extremal ray of type 5 on the variety X The image of EF’ in

X is acurve E ~ P', and, by the adjunction formula, o*Kx = Kx: + 5P”

Hence Kx - EH = 5- One can show that FE generates in X an extremal ray,

and F is the only irreducible curve belonging to this extremal ray The small contraction of this extremal ray leads to the initial variety Y which is not Q-Gorenstein

Here the variety X is not Gorenstein This is the general fact: there are no small extremal contractions on a Gorenstein three-dimensional variety (Mori

(1988), Clemens-Kollár-Mori (1988))

Remark 1.4.12 This example shows that it is rather difficult to construct 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 (Kolldr-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), Kollár (1989), Kollár (1991)) Let

X be a Q-factorial projective three-dimensional variety with at most termi- nal singularities, and let D € Div(X) A birational map y: X - - + X’ is called

a D-flop if there exists the following commutative diagram:

N4

where f and f' are birational morphisms (x = 1 © Ƒ) such thai:

(i) X! is a Q-factorial projective three-dimensional variety with at most ter- minal singularities;

(ii) let EB C X and E’ c X' be the exceptional loci for f and f’ respectively;

then codim E > 2, codim E’ > 2, and x: X — E 3 X'— E' is an isomor- phism;

(iii) the morphisms f and f’ cannot be decomposed into compositions X —

YY and X' = Y, > Y in such a way that conditions (i) and (i) for

Yì would be satisfied;

(iv) if D’ = y.(D) is the proper transform of the divisor D, then —D is relatively f-ample, and D’ is relatively f’-ample

Consider now the following simplest

Example 1.4.14 Let Z be a nonsingular three-dimensional complete va-

riety containing a surface F ~ P! x P! with normal sheaf Npyz ~

Op1xp1(—1,—1) Let Y be the variety obtained from Z by contracting F into

an ordinary double singularity Assume that in the commutative diagram

§2.1 Definitions, Examples and Simplest Properties 23

the varieties XY and _X’ obtained by contracting F’ onto nonsingular curves C

X and E’ c X’ along two pencils of lines are projective Then y; X +X’

is a flop transformation

Theorem 1.4.15 (Reid (1983a), Kollár (1989), Kolár (1991)) Let

f:.X —Y, D and E be as in Definition 1.4.13 Then:

(i) a D-flop exists:

(ii) every sequence yi: X; +X} of Dj-flops is finite, where Di = (xi-1)»Di_1 is the proper transform of D,-1, 1 = 1,2, ;

(iii) if X is nonsingular, then X‘ is also nonsingular

Remarks 1.4.16 (i) The proof of Theorem 1.4.15 in Kollar (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, because Kx -E = Ky - &' =0

(iv) Flops are applied extensively in three-dimensional birational geometry

For example, a minimal model of a three-dimensional variety X of general type

with Q-factorial singularities in general is not uniquely determined, and any two of them are linked by a birational transformation which factors into a composition of flops (Kolldr (1989))

Chapter 2 Basic Properties of Fano Varieties

82.1 Definitions, Examples and the Simplest Properties

Definition 2.1.1 A smooth projective variety X is called a Fano variety

if its anticanonical divisor —Kyx is ample If a normal projective variety X has singular points (for example, terminal, canonical, etc.), and some positive

integral multiple —-nKx, n € N, of an anticanonical Weil divisor —Kyx ts

an ample Cartier divisor (that is, in particular X is Q-Gorenstein), then X

is called a singular Fano variety A Fano variety with terminal Q-factorial singularities and p = 1 is called a Q-Fano variety Finally, a log terminal pair (X, B) is called a log Fano variety if a Q-Cartier divisor Kx + B is ample (that is, some its positive multiple being a Cartier divisor is ample) If B = 0, then X is simply said to be a log Fano variety

Proposition 2.1.2 Let X be a log Fano variety, and let f:¥Y — X be a

resolution of singularities Then:

(i) H'(X, Ox) = Hy, Oy) = 0, Vi > 0;

(ii) Pic(X) and Pic(Y) are finitely generated torsion-free Z-modules; Pic(X) ~ H?(X,Z), and Pic(Y) ~ H?(Y,Z);

(iii) numerical equivalence on the set of Cartier divisors on X, respectively on

Y, coincides with linear equivalence;

(iv) K(Y) = —oo

Assertion (i) follows from the Kawamata-Viehweg vanishing theorem (see 1.3.4) The second part of assertion (ii) follows from the cohomology sequences

for exact exponential sequences of sheaves over C

exp

0 — Z— Ox -— O%F — 0, 0—-+Z— 3 Oy “3 0% — 0,

assertion (i) and the isomorphisms Pic(X) ~ At(X,O%), Pic(Y) ~

H'(Y, OF)

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 Riemann—Roch theorem and the Kawamata-Viehweg vanishing theorem we get H°(X,Ox(D)) = H°(X,Ox) = 1, H°(Y,Oy(D’)) = H°(Y, Oy) = 1 Since X and Y are projective, it follows from the above that D = 0 and

D' = 0, and that Pic(X) and Pic(Y) are torsion-free

Corollary 2.1.3 Let X be a log del Pezzo surface Then X is rational

See (Kollar (1996a)) for generalizations of 2.1.2 and 2.1.3 to positive char-

acteristic

Remarks 2.1.4 (i) Corollary 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 Lemma 1.1 from (Kawamata (1988)), it follows that

the class group of Q-Weil divisors Cl & Q is finitely generated But unlike the case of Pic(X), the group Cl(X) can have torsion One can find such examples

among those three-dimensional varieties X C P’ whose hyperplane sections

are Enriques surfaces (see Conte (1982), Conte-Murre (1985))

In view of 2.1.2 (ii), for a log Fano variety X there exists the greatest

rational r = r(X) > 0 such that —Ky = rH for some (ample) Cartier

divisor H (the equality here means the equality of elements of the group Pic(X) @ Q); such an r = r(X) is called the index of the Fano variety X, and the divisor H (respectively the linear system ||) is called a fundamental

divisor (respectively the fundamental system ) on X We shall call the self-

intersection index d = d(X) = H4™* the degree of the Fano variety X (if

H = —Kyx, and —Kyx is very ample, then d is the usual degree of X with

respect to the anticanonical embedding y\_ Kx |: X => pdm ¬ We remark

that if X is Gorenstein, then the index r = r({X) is a positive integer

§2.1 Definitions, Examples and Simplest Properties 25

Examples 2.1.5 (i) In dimension 1 there is a unique Fano variety up to isomorphism namely P! Two-dimensional Fano varieties are called del Pezzo surfaces The following is the complete list of them: P?; a smooth quadric

Q Cc P®; a geometrically ruled surface F,; with the exceptional section s,

s* = —l, a series of surfaces F = Fy Cc P¢ of degree d = K}, 3 < d < 7;

a double cover F2 of P? ramified along a smooth curve of degree 4; a double cover F; of a quadric cone Q’ C FP? ramified along a smooth curve of degree 6 not passing through the vertex of the cone Surfaces Fj can be obtained by blowing up of 9 — d points on P? which are in general position, In particular, all del Pezzo surfaces are rational (see Manin (1972), Nagata (1960) and also

2.1.3)

(ii) An analogous classification exists for del Pezzo surfaces with canoni-

cal (i.e Du Val) singularities (see Du Val (1934), Demazure (1980), Brenton

(1980), Hidaka-Watanabe (1981)) If X is a singular del Pezzo surface with

Du Val singularities, and (—Kx)* = d, then X is one of the following: 1) d=8, X =Q’ Cc P® is a quadric cone;

2) 3<d<6, X = XqC Pt is a projectively normal surface of degree d; 3) d = 2, X can be represented as a double cover X — P? ramified along a singular curve of degree 4;

4) d=1, and X can be represented as a double cover X — Q’ of a quadric

cone ramified along a singular curve cut out on Q’ by a surface of degree 3

A minimal resolution X of the surface X in all the cases but 1) is a blow-up of P? at 9—d points which are in almost general position; see Demazure (1980)

(iii) Let G C PGLa(k) be a finite group, and S = P?/G Then X is a log

del 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 del Pezzo surface with p = 1 is of the form P?/G (see Alexeev- Kolpakov (1988)) Log del Pezzo surfaces have been studied by V A Alekseev

and V V Nikulin The complete classification of log del Pezzo surfaces with

indices of singular points < 2 was obtained in Alexeev-Nikulin (1989) The

set of possible values of indices of log del Pezzo surfaces was described in Alexeev (1988) The questions concerning the boundedness of the degrees

of such surfaces in terms of multiplicities and indices of singularities were considered in Nikulin (1989a), Nikulin (1988), Nikulin (1989), Alexeev (1988) (iv) Now we give some of the simplest examples of Fano varieties of di-

mension n = dim X > 3 First of all, these are projective spaces P”, smooth

hypersurfaces Xq C P+! of degree d < n+ 1, smooth complete intersections

Xd ccd, C P’ of hypersurfaces of degrees d,, ,d, such that od, <N

Let Y be a Fano variety of index r Let H be a fundamental divisor on Y (that is, -Ky = rf), and let 7: X -+ Y be a cyclic cover of degree m with smooth ramification divisor B C Y If B ~ bH, where b < rm, then it follows

from the Hurwitz formula Ky = 1*(Ky + +B) that X is a Fano variety of index r’ = r — © (note that m divides 6)

A more general construction (see Mori (1975), Dolgachev (1982)) Let P =

P(eg, ,e) be a weighted projective space, that is, P = ProjClxo, ,2n], where degx; = e; Let X C P be a smooth complete intersection of hyper- surfaces of degrees d,, ,dm which does not intersect the set Uzs15,%, where S; C P is the closed subset determined by the ideal generated by {z; | k fe;}

Then the restriction of the sheaf Op(1) to X is an invertible sheaf Ox (1), and

if dim X = N—m > 3, we have Pic(X) = Z-Ox(1) The canonical class of X

can be calculated by the formula Ox (Kx) = Ox(3> di — ¥° e;) From this we

find that if Sod; < S*e;, then X is a Fano variety of index r = Soe, — Sod;

with p= 1

(v) Grassmannians G = Gr(m, N) of m-dimensional subspaces in an N-

dimensional linear space are also Fano varieties, and —Kg = N-H, where H

is a hyperplane section of G under the Pliicker embedding G = Gr(m, N) P(A™C’), From the adjunction formula, it follows that smooth complete in- tersections of G with hypersurfaces of small degrees are also Fano varieties

In particular, a general section of Gr(2,5) C P® by a subspace of codimen- sion 3 or by a subspace of codimension 2 and a quadric are three-dimensional

Fano varieties X5; C P® and Xiq C P” with p = 1, and —Kx, = Ox,(2),

—K xy, = Ox,,(1)

(vi) Q-factorial Fano varieties naturally arise in the Mori Minimal Model Program Many examples of such varieties can be found among toric varieties (sce below) Other examples of singular (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 terminal singularities; see Clemens-

Kollar-Mori (1988)) For example, let G C PGL,41(k) be a finite subgroup;

then P"/G is a log Fano variety with p = 1 and Q-factorial singularities

If G c PGL4(k) is a subgroup of order 2, then the following two cases are

possible:

a) G is generated by the element g = Diag(1,1,1,—1); in this case P?/G is isomorphic to the cone over the Veronese surface and is a Q-Fano variety (see Example 1.1.1);

b) G is generated by the element g = Diag(1,1,—1,—1); in this case X =

P” /G is isomorphic to the complete intersection of two quadrics xz = y?

and tv = u? in P® The variety X has canonical Gorenstein singularities along two conics which are cut out by the planes 7 = y = z = 0 and

u=v=t=0

(vii) If on a smooth projective variety X, an anticanonical divisor —Kx

is nef and big, then, by Theorem 1.3.6, the linear system | — mK x] is free for m > 0 From this it follows that the anticanonical ring R(—Kyx) =

® H°(X,Ox(—mKx)) is finitely generated Then the anticanonical model m20

X_can ‘= Proj R(—K x) is birational to X (there exists a natural birational

isomorphism X — X_can) and is a Fano variety with at most canonical sin- gularities.

§2.1 Definitions, Examples and Simplest Properties 27

(viii) (T Fujita) Let X be a log Fano variety, and let —K = rH for some

ample Cartier divisor H on X We set E = Ox 6 Ox(H) The linear system

|Ope)(m)| where Ozie)(1) is the tautological linear bundle on P(€), is base

point free for m >> 0 and determines the morphism P(€) > X' c P% which

contracts the exceptional section E It is easy to check that

Kee) = fi Kx: + (r — LE

From this we get that X’ is a log Fano variety of dimension dim X + 1 and index r +1 This variety is called a generalized cone over X For example, let X = P*, Ox(H) = Op(d), and r = 4 Then X is a usual cone over the surface Sy2z which is the image of P? under the embedding determined by the linear system |Op2(d)| If d = 1, then X' ~ P*; if d= 2, then X’ is the cone over the Veronese surface (see the example above); if d = 3, then X‘ is a Fano variety with Gorenstein canonical singularities; if d > 4, then X’ is a log Fano

variety We remark that the last examples can also be obtained as quotients

of P? by actions of abelian groups

The natural generalizations of Grassmannians (see Example 2.1.5 (v)) are homogeneous spaces Consider a semisimple linear algebraic group G and its parabolic subgroup P C G Then the quotient X = G/P is a projective

algebraic variety which is a homogeneous space with respect to the action

of the group G Now we need to introduce some notation Let T C P be a maximal torus, and & := Hom(7T.C*) @R Let A C FE be the root system corresponding to G Let 2 Cc A be a basis of the root system A (that is, a sct of simple roots), and let W = Ng(T)/T be the Weyl group

We can assume that P = P(Q) is a standard parabolic subgroup, that is,

P is generated by some fixed Borel subgroup and the group We, where We

is the subgroup of W generated by reflections with respect to the elements

of O The cohomology ring H*(X,Z) and the total Chern class c(X) of the variety X = G/P can be expressed in terms of the root system A (see Borel (1953), Borel-Hirzebruch (1958)) Here we restrict ourselves to the description

of H?(X,Z) and ¢(X) = —Kx

Proposition 2.1.6 (Borel (1953), Borel-Hirzebruch (1958))

(i) There exists a canonical injective homomorphism of Z-modules

v: H?(X,Z) > E The image of v is orthogonal (with respect to the W-

invariant bilinear form on F7) to the elements a;, i € O, and is generated

by the fundamental weights w;, i ¢ O

(ii) c1(X) = So ai, where a; runs over the set of positive roots complementary

to @, that is, positive roots a, such that in the decomposition of a; with respect to the basis 3/ at least one element from X\© occurs with a strictly positive coefficient

(iii) An element b © H?(X,Z) is positive if and only if (b,a;) > 0 for all

M

Corollary 2.1.7 (Borel-Hirzebruch (1958), Kollar (1996a)) On the var- ety X = G/P, the anticanonical divisor —Kx is ample, that is, X is a Fano

variety

It is also possible to describe the Mori cone NE(X) and extremal rays on

X in terms of subgroups of G: if P’ > P is another parabolic subgroup of

G, then the canonical morphism X = G/P — G/P’ is a contraction of an extremal 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 variety of flags in P"~? Example 2.1.8 Let G = SL3(C), and let P = B be a Borel subgroup (the subgroup of upper triangular matrices) Then X = G/B is the variety of complete flags in P?, and dim X = 3 The root system A is of type Ag (see

Fig 2.1.8), © = {a1,a2}, 0 = 0 The group H?(X,Z) is generated by the

a2 a1 + Q2

Fig 1

fundamental weights w, = Zay + saa, 1Ua — sai + 502! therefore p(X) = 2 Calculate now the index r of the Fano variety X The class of an anticanon-

ical divisor —K’x is the sum of all positive roots:

ci(X) = a, +Qo+ (ay + a2) = 2(ay + Ø2) = 2( + We) :

Thus we get that r = 2 Since p(X) = 2, then X has exactly two extremal rays, and their contractions y;: X — P? are P'-bundles (that is, are of type

C2) The variety X can also be realized as a divisor of bidegree (1,1) on

P2 x2 or as the projeetivization Pp2(Tp2) of the tangent bundle Tp2 of P?

(see 3.3.1)

Consider now homogeneous varieties X = G/P with Picard number

p(X) = 1 In this case the parabolic subgroup P should be maximal, and the corresponding subset O C 5 is obtained from X' by removing one ele- ment Therefore we can assume the group G is simple We get the following

examples.

§2.1 Definitions, Examples and Simplest Properties 29

Examples 2.1.9 (Borel-Hirzebruch (1958)) (i) G = SL,»,(C) Up to conju-

gation, G contains exactly m— 1 maximal parabolic subgroups P;, , Pm_1- The quotient varieties G/P, are isomorphic to the Grassmannians Gr(k,m)

(ii) G = SO,,(C) For every maximal parabolic subgroup P C G, the

quotient variety G/P is isomorphic to the isotropic Grassmannian, that is,

to the subvariety of Gr(k,m) consisting of k-dimensional linear subspaces

Vc C” such that g(V, V) = 0, where q = (-,-) is a non-degenerate symmetric bilinear form on C™ Consider now the cases of small m

a) m= 5 The root system A is of type Bo:

ESO ,

ont œ2

One can assume that P = P, = Po,, where O; = A-— {a;} We get two varieties: G/P, which is isomorphic to a smooth three-dimensional quadric Q® C P*, and G/P2 which parametrizes lines lying on a smooth three-dimensional quadric (it is known that it is isomorphic to P*) b) m= 7 The root system A is of type Bs:

A(ay + 2a + 2a) = Awe ›

that is, G/P2 is a 7-dimensional Fano variety of index 4 Likewise, we get

that G/P3 is a 6-dimensional Fano variety of index 6 (and therefore G'/P;

is isomorphic to a quadric Q° C P’, see 3.1.15)

(iii) G = Sp,,,(C) If P C G is a maximal parabolic subgroup, then G/P is

isomorphic to the isotropic Grassmannian with respect to a non-degenerate alternating bilinear form

(iv) G is a group of type Gy There exist exactly two maximal parabolic subgroups P, and P, in G The quotient variety G/P, is isomorphic to a 5- dimensional quadric Q° Cc P®, and G/P, is a 5-dimensional Fano variety of index 3 (see Mukai (1989))

A similar description can also be obtained for G/P, where G is a group of type Fy, Eg, 7 or Eg

Another convenient way to construct many examples of Fano varictics is

given by toric geometry (see Kempf et al (1973), Danilov (1975), Oda (1978))

Definition A variety X is called toric if there exists an action T : X of

an algebraic torus T on X by bireqular automorphisms which has a dense (in the Zariski topology) orbit U C X equivariantly isomorphic to T

Let AY = Z” Let N = M* be the dual Z-module, and let Ne = N &

R Elements m € Af, m = (m1, ,™mn), correspond to monomials 2” =

xy ai € C(M] = Clx, , 2] To every cone o C Ng (we assume that

all the cones are determined by a finite number of inequalities of type m > 0, where m € Mf &R), one can associate a finitely generated ring C[éM A/] and, therefore, an affine toric variety X,- = SpecC[é 9 M)

A fan F is defined to be a finite set of cones a C Ng such that the following

three conditions hold:

1) ifo€ Fand7 € F, theno M7 € F;

2) ifo € F and r is a face of o, then 7 € F;

3) every cone o € F has a vertex

For every fan F’, there is a toric variety Xp associated to it; Xp is covered

by affine toric varieties X,,, where o € F, and every two varieties X, and X/

are glued along X, if r is a common face of ø and a’ It is known that every

toric variety can be obtained in this way We consider here only complete varieties Xp This condition is satistied if and only if the fan F’ is complete, that is, Ne = LU o (see Danilov (1975)) All the information concerning the

ơcF

variety Xp can be expressed in terms of the lattice N and the fan F For

example, the Picard group Pic( Xr) is isomorphic to the quotient group of the group of piecewise linear functions on Ng having integer values on N by the

subgroup M (see Danilov (1975), Oda (1978))

Examples 2.1.10 (i) Let M = Z", N = M* Let e1, ,¢€n bea basis in N,

and ey = —€] — — €» Consider the fan F' consisting of cones (@;, ,€:,), k<n,0<12; <n Then the variety Xp is isomorphic to P” It is known that

every smooth complete toric variety with p= 1 is isomorphic to P”

(ii) Toric del Pezzo surfaces are exactly the following ones: P?, P! x P!, F,,

Fe, Fy (see Example 2.1.5 (i))

(iii) The complete classification of three-dimensional toric Fano varieties

was obtained in papers of Batyrev (1981), Watanabe-Watanabe (1982) (see Table 12.8, Chap 12) Every such variety Xj} is determined by a fan F C

Ng which in its turn is determined by a Z?-weighted triangulation of the two-dimensional sphere $* C R* = Ng (every vertex of the triangulation corresponds to a ray in F and its Z?-weight corresponds to coordinates of a primitive vector from N lying on this ray) There is an upper bound p < 5 for the Picard number of three-dimensional toric Fano varieties In the case

p = 2 the triangulation of S? Cc R? is determined by the graph represented

§2.1 Definitions Examples and Simplest Properties 31

Vì V5 = {oo}

Fig 2

on Fig 2.1.10, and Z°-weights are determined by the matrix N for which one

has only the following possibilities:

is a the blow-up of a point on P3, r = 2, d = 7

Remarks 2.1.11 (i) The list of four-dimensional smooth toric Fano vari-

eties can be found in Batyrev (1984) and Batyrev (1998)

(ii) Voskresenskii-Klyachko (1984) completely classified all smooth toric

Fano varieties X which possess the following property: the involution 7: T —

T, t — —t of the torus T can be extended to a biregular involution of the variety X

(iii) Ried (1983c) showed that the Minimal Model Program holds for toric varieties of any dimension with terminal Q-factorial singularities (see also

Danilov (1982))

(iv) Toric geometry can be used to construct a great number of examples

of singular Fano varieties For example, weighted projective spaces and any

quotients of P” modulo abelian groups are toric Fano varieties A A Borisov and L A Borisov obtained the following result: every three-dimensional toric Q-Fano variety is either a weighted projective space of one of the follow-

ing types P(1,1,1,1) = P®, P(2,1,1,1) (the cone over the Veronese surface),

P(8, 2,1, 1), P(5, 3, 2,1), P(5,4,3, 1), P(7,5, 3,2), P(7, 5,4, 3), or a toric variety determined by the simplex in N with vertices at points (1,0,1), (—2,1,1),

(1, —2,0), (0, 1,2)

Next we discuss the simplest properties of Fano varieties

The proposition below follows immediately from the Kawamata—Viehweg vanishing theorem

Proposition 2.1.12 Let X be a n-dimensional log Fano variety of in- dex r, and H be a fundamental divisor Then

H'(X,Ox(mH))=0, Vi>0, VYm >—r

Corollary 2.1.13 (Shokurov (1985)) The index of a log Fano variety X does not exceed dim X +1

Proof By the Riemann Roch theorem, y(Ox(m#)) is a polynomial of degree dim X By Proposition 2.1.12, the roots of this polynomial are integers

n=-—1,-2, ,1— fr], whence r < dim X +1

Similarly, using duality, we obtain

Corollary 2.1.14 (see, for example, Iskovskikh (1979a)) Let X be an n-

dimensional log Fano variety of index r Let H be a fundamental divisor on

X, and let d= H” be the degree of X Then:

(i) ofr >n— 2, then

H°(X, Ox(H)) = sdứ —m+3)+w~1,

andd=1 forr=n+1,d=2 forr=n;

(ii) ifr =n—2, and X has at most canonical Gorenstein singularities, then

H°(X,Ox(H)) =g+n—1,

where g = 5 +1> 2 is an integer

The integer g = g(X) = He 41 is called the genus of an n-dimensional Fano variety X of index n — 2 It can also be defined for a three-dimensional Fano variety X with Gorenstein singularities of arbitrary index to be Kx)" +1 The reason for this definition is the fact that for a three-dimensional (non- singular) Fano variety X with very ample —K x, the image yj_«,\(X) = Xoy—2 © P9t! 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 true, that is, every smooth three-dimensional variety X2,—2 in Pet! with canoni-

cal curve-sections is a Fano variety (see Conte-Murre (1986)) In the original papers by G Fano, this property was taken to be the definition of the class

of varieties under investigation

The ample sheaf Ox (H) is not necessarily generated by its sections, that is,

the corresponding mapping Yj 4): X ~ ~ + Pdim |) is not necessarily a morphism

(see Sect 2.3) For three-dimensional Fano varieties X with y)y) being a morphism, we have the following

Proposition 2.1.15 Let X be a three-dimensional Fano variety of index r with canonical Gorenstein singularities, and let -—Ky = rH, where H 1s a fundamental divisor on X Assume that the linear system |H| is base point free and determines a morphism y\y\: X — #Imj(X) C pdim|HỊ, Then deg #|H| =

1 or 2

Proof Let d = H® Substituting formulas 2.1.14 into the inequality

deg yj y\(X) = codim yiy\(X) +1, which is obviously true for any subvariety of a projective space not lying in a

hyperplane (see Corollary 3.1.4), we obtain the inequalities:

(1988))

Theorem 2.1.16 (Iskovskikh (1979a)) Let X be a nonsingular three-

dimensional Fano variety of index r and genus g Assume that the anticanon-

ical linear system | — Kx| determines a morphism y:X — X' Cc P#t! which

is not an embedding Then yp: X — X' is a double cover with a smooth ramt- fication divisor D C X' The variety X is completely determined by the pair (X’, D), and for this pair only the following cases are possible:

(i) X’ C P® is a cone over the Veronese surface, and D C X' is cut out

by a cubic hypersurface; in this case X is isomorphic to the variety from

3.1.6 (i);

(ii) X' = P° and D C P® is a surface of degree 6; in this caser = 1, g(X) = 2,

and Pic(X) = Z;

(iii) X’ = Q CP is a smooth quadric, and D C Q is cut out by a quadric in

P+; in this case r = 1, g(X) =3, and Pic(X) = Z;

(iv) X’ = P! x P? c P® embedded by Segre, and D is a divisor of bidegree (2,4); in this case r= 1, g(X) = 4, and Pic(X) = ZZ;

(v) X! = Pp(€), where € = Op (2) 6 Op: (1) GB Op (1), and X’ is embedded in P® by the linear system |Ope)(1)|, and D € |Opre)(4)|; in this case r = 1, g(X) = 5, and Pic(X) = Z® Z; the variety X can also be realized as a

blow-up of a Fano variety Y2 of index 2 along a nonsingular elliptic curve

Hy, 0 He, where Hi, Ha € 2| — Ky,|;

(vi) X’ =P! x P? c P® embedded by the linear system |p}Op: (2) & p3Op2(1)|, and D € |p}Op2(4)|; in this case r = 1, g(X) = 7, and Pic(X) = 279:

the variety X is isomorphic to P! x F, where F is a del Pezzo surface of

degree 2

Corollary 2.1.17 (cf Iskovskikh (1979a}) Let X be an n-dimensional

nonsingular Fano variety of index r = n — 2 and genus g Assume that

the fundamental linear system | — 4K x| = |H| determines a morphism

@:X — X'’ c Py"? which is not an embedding Then y:X — X' is a double cover with a smooth ramification divisor D Cc X' The variety X is completely determined by the pair (X’, D) Ifn > 4, one has for this pair only the following possibilities

(i) X’ =P", and D is a hypersurface of degree 6 In this case g(X) = 2, and Pic(X) = Z

(ii) X’=QCP"*! is a quadric, and D C X' is cut out by a hypersurface of degree 4 In this case g(X) = 3, and Pic{X) = Z

(iii) X’ =P! x P? cP” embedded by Segre, and D € |pšps(4)| In this case

n = 4, g(X) = 5, and Pic(X) = Z@ Z The variety X is isomorphic

to P! x Yo, where Yo is a three-dimensional Fano variety of index 2 and

degree 2

82.2 Some General Results

Definition 2.2.1 A normal variety Y C P’ which does not lie in a hy- perplane (it is often said to be non-degenerate) is called projectively normal if one of the following clearly equivalent conditions is satisfied:

i) for any integer m > 1, every divisor from the complete linear system y Ụ Ọ Oy(m) ơn Y is cut out by some hypersurface of degree m in PX;

(ii) the natural homomorphism of graded algebras

§2.2 Sone General Results 35

is surjective, where S* H(.) is the symmetric algebra generated by the space H"(-)

Definition 2.2.2 A variety Y C P% is called fine if it is projectively

normal and h'(Oy(m)) = 0 for allt, 0<i<dimyY

Lemma 2.2.3 Let Y be an irreducible complete variety Let L be an in- vertible sheaf on Y Let s © H°(Y,L) be a non-zero section, and let DC Y

be the divisor of zeros of the section s Assume that:

(i) the natural homomorphism H°(Y,L) + H°(D, L|D) is surjective;

(ii) the homomorphisms S™~! H®(Y, L) > H(Y, LD) are surjective for

m > 2, where S"~! H°(-) denotes the (im — 1)-st symmetric power of H°(-) Let I(Y¥,L) be the homogeneous ideal which is the kernel of the

homomorphism S* H°(Y.L) + @msoH°(¥Y,L°™), and let Tu(Y.L) C I(Y.L) be its homogeneous part of degree m Similarly we define I(D, L|D)

and I,,(D,E|D)

Then for any element F,, C Im(D,L|D), there exists an element FE, € Im(¥, L) such that F,|D = Fy, that is, the homomorphism On: Im(¥,L) > Tin(D, L|D) is surjective; moreover, Fy (form = 2) is uniquely determined by the element Fy (Note that the conditions of the lemma are obviously satisfied for projectively normal varieties Y CPN and L = Oy(1) with any m > 0) Proof Consider the following commutative diagram:

where the exactness of the first column is precisely condition (ii), and the

exactness of the second row is condition (i) of the lemma The desired sur- jectivity of the homomorphism a,,, can be proved by the standard diagram method (the snake lemma) The uniqueness of F2 is obvious

Corollary 2.2.4 (see, for example, Iskovskikh (1977)) Under the assumptions of Lemma 2.2.3 for all m > 2 (for evample, for projectively normal Y c PN), if I(D) is generated by elements of degree < r for some integer r, then I(Y) is also generated by elements of degree < r In particular,

if D C PN”! is an intersection of quadrics containing D, then Y is also an intersection of quadrics containing Y

Lemma 2.2.5 In the preceding notation, assume that the restriction se-

H°(Y, Le") 5 H°(D, (L)D)°") 0

is exact for allm > 1 Then if the graded algebra @m>vH(D, (L|D)®™) ts

generated by elements of degree <r for some integer r, then the graded algebra Dm>o0H(Y, L®™) is also generated by elements of degree <r

This is proved by induction as well as the preceding corollary

Lemma 2.2.6 (Iskovskikh (1979a)) Let Y be a nonsingular curve of

genus g, and let L = Oy(D) be an invertible sheaf of degree deg L > 2g +1

Then the sequence

0 — 1(Y,L) 3 S* H°(y,L) 4 @ H°(y,L°”") 50

m>0

as 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 P? is a cubic curve

Lemma 2.2.7 (Iskovskikh (1979a)) Let L = O(D) be an ample invertible

sheaf on a smooth variety Y Assume that the map

gm HY, L) _—> AY, pe)

is surjective for allm > 1 Then divisor D is very ample (and the image

#Ip,Ÿ) C IPX its obviously projectively normal)

Proof Consider the following commutative diagram:

PimD| P(H°Œ, 13) a

XS

ep} và P(H"(Y, L)) 0 ai

Here yp, is an embedding for some m > 0: a is an embedding by the

assumption of the lemma that the map is surjective; ( is the standard Veronese embedding From this it follows that yp) is also an embedding

Corollary 2.2.8 Under the conditions of Lemma 2.2.6, the sheaf L =

Oy(D) is very ample The image ~\p|(Y) is a projectively normal curve and

is an intersection of quadrics containing it

We now formulate the classical Noether-Enriques—Petri theorem on canon-

ical curves.

82.2 Some General Results 37

Theorem 2.2.9 (see, for example, Griffiths-Harris (1978)) Let

Y Cc P9! be a smooth canonical curve of genus g > 3 Then:

(i) Y is projectively normal in P9~!;

(ii) if g = 3, then Y 1s a plane curve of degree 4; if g > 4, then the graded ideal Iy is generated by components of degree 2 and 3;

(iit) the ideal fy is generated by the component of degree 2 in all the cases except for the following ones:

a) Y is a trigonal curve, that is, it has a one-dimensional linear system

gh of degree 3;

b) Y ts a curve of genus 6 isomorphic to a plane curve of degree 5;

(iv) in the exceptional cases a) and b), the quadrics passing through Y c P97} cut out a surface F C P9~! which is respectively:

a) a nonsingular geometrically ruled surface of degree g — 2 in P9~! for

g > 5, and a quadric (perhaps singular) in P’ for g = 4;

b) the Veronese surface Fy in P°

In the sequel we need to know some properties of non-degenerate projective varieties Y c P*% with the property

deg Y = codimY + 1 (such varieties are in some sense extremal since we always have that deg Y >

codim Y + 1, see Corollary 3.1.4)

Let dj > dg > > dy, > 0 be a set of integers, and let E = Om (di) © D Opi (dy,) be a locally free sheaf of rank m on P! We set Y = P(E) (the projectivization of the bundle € on P'), and let 7:Y — P! be the natural projection Let £ = m*Opi(1), and let M = Oy,pi(1) be the Grothendieck tautological invertible sheaf (that is, 7,47 = €) Let L and AMZ be divisors on

¥Y such that £ = Oy(L) and M = Oy(M)

Proposition 2.2.10 In the preceding notation, we have:

(i) the sheaf M is generated by its sections; M is ample if and only if a; > 0,

Vì — Ì, ,Tn;

() the tmage @A4(Y ) is a fine variety;

(iii) h°(M) = ROM) = h(E) = (di + 1);

¿=1

(iv) the ideal Iy, the kernel of the natural homomorphism a: S* H°(M) =

@n>0H?(M®"), either is trivial or is generated by elements of degree 2; (v) deg yau(Y) = codimY + 1;

(vi) let 7 be the least index (1 < j < m) such that d, = 0; then yu is

a cone with verter pu(P(G7L,Op:)) ~ P2 and base isomorphic to

P(@!7{Om (di)); the restriction pm |P(@r Op) coincides with the pro-

jection py: P™~4 x Pp! (prs,

Let Y = P(€) be such as in 2.2.10 Then the variety px,(Y) is called the

scroll of the vector bundle € if it is nonsingular and a cone over the scroll of the vector bundle E if it is singular

Now we come to the classical result we are interested in (see Fujita (1975),

Iskovskikh (1979a), Eisenbud-Harris (1987))

Theorem 2.2.11 (F Enriques) Let Y c P* be a non-degenerate variety satisfying the condition

deg Y = codimY +1

‘Then Y is one of the following:

(i) @ projective space P* ;

(ii) a quadric in PX ;

(iit) @ scroll of a vector bundle;

(iv) a cone over a scroll of a vector bundle;

(v) a Veronese surface Fy c P°;

(vi) @ cone over the Veronese surface

Below we describe some numerical and geometrical properties of P(€)

Proposition 2.2.12 In the preceding notation, we have:

(i) L2=0, Le M1 =1, M™ =r d;;

2=]

(ii) Pie(Y¥) =Z-L+Z-M, and Ky = —mM + (Š) dị — 9)

¡=1

Proposition 2.2.13 Assume in addition that dy > d, For every integer

c, dy >> dm, we set Y, = P(E.), where E, >= Da,<cOp: (dj), and identify Y, with its image in Y with respect to the embedding determined by the projection

E — €, Let a and b be such integers that H°(M®* @ L®>) £0 Then every section s © H®(M®4 @ £®°) has zeros of order > q on Y, if and only if the wnequality

øe + b + (dị — c)(g — 1) <0 holds

Proof For a > O there exist the following natural isomorphisms: H°(MẺ* @ £S)) ~ H(m,.(M®" ~ £8)) ~ H°(P!,S*E & Opi(b)), where S* € is the a-th symmetric power of the sheaf € By assumption, H°(M®* @

£®!°) £ 0, therefore a > 0 Let €2 = @a,s-Opi (di); then £ = £/ @® E., and

SE = G2, S' E! @ S*-"E The subvariety Y C Y is determined by the

vanishing of all the sections from H°(P!,€!) c H®°(Y,M) Therefore every section s € H°(P!,S* € @ Opi(b)) has zeros of order > q on Y if and only if

s has degree > g in the terms from €/ 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* € S*~* €, ® Opi (b) into 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) be negative Thus, the desired inequality must hold