LOG MODELS OF BIRATIONALLY RIGID VARIETIES

by

Ivan Cheltsov

A dissertation submitted to the Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy

Baltimore, Maryland

March, 2000

Abstract.

The thesis contains a new method that allows to describe all birational maps of a given

birationally rigid variety into �brations, which general �ber has Kodaira dimension zero.

Developed method is e�ectively applied in the thesis to birationally rigid Fano 3-folds,

conic bundles and some higher dimensional Fano varieties.

Birational rigidity of an arbitrary smooth hypersurface of degree N in PN is proved in

the thesis in case N = 5; 6; 7 or 8.

The thesis gives an e�ective way of regularization of �nite subgroups of birational

automorphisms.

This work was partially supported by NSF Grant DMS-9800807.

Acknowledgments.

I wish to thank my advisor, Professor V.V.Shokurov, who was so generous with his

ideas, time and expertize.

I would like to express my gratitude to Professors V.A.Iskovskikh, A.V.Pukhlikov and

A.Corti, whose pioneer work has motivated this project from the beginning.

I want to thank T.Abe, M.Grinenko, J.Park, Y.Kachi, F.Ambro, L.Wotzlaw, A.Radtke

and G.Hein for useful conversations and helpful comments.

I am indebted to my parents, Anatoly Cheltsov and Nataly Cheltsova, and to my wife,

Elena Cheltsova, without whom the thesis would not exist.

I am very grateful to my son, Ivan Cheltsov, whose perfect behavior allowed me to �nish

the thesis.

I want to thank my friends S.Kasyanov, A.Pankratiev, A.Kovalyov and V.Chekunov for

being my friends.

Dedication.

I dedicate this dissertation to my parents, Anatoly Cheltsov and Nataly Cheltsova.

CONTENTS

Introduction.

x1. Historical background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

x2. Obtained results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

x3. Thesis' description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Chapter I. Log pairs on algebraic varieties.

x1. Movable log pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

x2. Global properties of log pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

x3. Local properties of log pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Chapter II. Birational rigidity.

x1. What is birational rigidity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

x2. Conditions and criteria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter III. Del Pezzo surfaces.

x1. Smooth del Pezzo surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

x2. Double cover of quadric cone in P3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

x3. Double cover of P2 rami�ed in quartic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

x4. Cubic in P3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter IV. Birationally rigid Fano 3-folds.

x1. Smooth Fano 3-folds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

x2. Double cover of P3 rami�ed in sextic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

x3. Quartic 3-fold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

x4. Double cover of quadric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

iv

Chapter V. Conic bundles.

x1. Surface conic bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

x2. 3-fold conic bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Chapter VI. Fano hypersurfaces.

x1. Hypersurfaces of degree N in PN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

x2. Quintic 4-fold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

x3. Sextic, septic and octic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Chapter VII. Singular Fano 3-folds.

x1. 3-fold with unique elliptic structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

x2. Singular quartic 3-fold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Appendix. Regularization of birational automorphisms.

x1. Explicit regularization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

x2. One example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

x3. Question of Yu.I.Manin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Vitae. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

v

INTRODUCTION

x1. Historical background.

Results of this thesis go back to classical papers [No] of M.N�other and [Fa] of G.Fano. In

the beginning of the 20-th century these two papers were not so closely related as it seems

now. The explanation of their relation to each other is a revolution in birational geometry

that was made in the 80-s by Y.Kawamata, S.Mori and V.V.Shokurov (see [KaMaMa],

[Mo] and [Sh1]).

New approach to birational geometry (Minimal Model Program) proved a number of

conjectures and gave new proofs of many classical results. Moreover, tremendous number

of apparently distinct problems became relevant from the new point of view.

M.N�other and G.Fano studied the group of birational automorphisms of P2 and smooth

quartic 3-fold in P4 correspondingly. In particular, M.N�other proved the following result.

Cremona Group. Bir(P2) is generated by projective automorphisms of P2 and Cre-

mona involution.

This theorem of M.N�other inspired many mathematicians to consider a similar problem

for P3. However, some methods of [N�o] work only for surfaces.

Despite the large time interval (almost half of a century) between papers [No] and [Fa],

their methods are very similar in general. Both papers contained gaps and paper [Fa] was

almost completely forgotten until the 70-th.

In 1966 Yu.I.Manin proved the following theorem (see [Ma1-2]).

Del Pezzo Surfaces. Let S be smooth del Pezzo surface over an arbitrary algebraically

non-closed �eld, such that Pic(S) �= Z and

S �=

8>><>>:double cover of the quadric cone in P3 rami�ed in quartic;

double cover of P2 rami�ed in smooth quartic curve;

cubic in P3:

Then the group Bir(S) is generated by Aut(S) and so-called Bertini and Geizer involutions.

1

Birational automorphisms attracted attention of the mathematical community in 1971

due to the following result of V.A.Iskovskikh and Yu.I.Manin (see [IsMa]).

Quartic Theorem. Let X be a smooth quartic 3-fold in P4. Then

Bir(X) = Aut(X):

In particular, Quartic Theorem implied that every smooth quartic 3-fold is non-rational,

because the group of automorphisms of smooth quartic 3-fold is �nite. On the other hand,

some special quartic 3-folds were known to be unirational.

B. Segre's Example. Quartic

x40 + x0x34 + x41 � 6x21x

22 + x42 + x43 + x33x4 = 0

is unirational.

So, Quartic Theorem gave the �rst counterexample to the very famous L�uroth problem

for 3-folds. New epoch of birational geometry began.

At almost the same time methods of [IsMa] were applied to a wide class of algebraic

varieties (see [Is3]). The main part of this class were smooth 3-folds with ample antica-

nonical divisors, because these varieties look like rational ones, but nevertheless some of

them are not rational.

In the 80-s the application area of the methods of paper [IsMa] was extended in such a

way that it covered conic bundles. Namely, V.G.Sarkisov obtained the following condition

on birational rigidity of conic bundles (see [Sa1-2]).

Conic Bundles. Let � : X ! S be \very rami�ed"1 standard conic bundle 3-fold.

Then the following exact sequence of groups exists

1! Bir(X�)! Bir(X)! Bir(S);

where X� is a geometric generic �ber of morphism �.

Note that V.G.Sarkisov used his birational rigidity condition to construct non-rational

3-fold with trivial intermediate Jacobian, which contradicted prevailing opinion that a

1j4KS +Dj 6= ; for rami�cation divisor D of conic bundle �.

2

3-fold is non-rational if and only if its intermediate Jacobian has a non-trivial GriÆths'

component. Inspired by V.G.Sarkisov's results, V.A.Iskovskikh formulated and studied in

papers [Is4-5] the Rationality Criterion, which is almost proved nowadays.

Ten years later, generalizing V.G.Sarkisov's ideas A.Corti announced in [Co1] so-called

Sarkisov Program, which became very famous for the next ten years.

One should probably note here that geometry of three-dimensional conic bundles was

thoroughly studied since the beginning of the 70-s using the method of intermediate

Jacobian (see [Tu]). As an application non-rationality of a smooth cubic 3-fold was

proved. Nevertheless, these methods could not be extended in higher dimensions as easily

as methods of V.A.Iskovskikh and Yu.I.Manin (see [Ha]).

Minimal Model Program shed new light on all papers on the structure of the group of

birational automorphisms. Surprisingly, even a notion of \the birationally rigid variety"

had been changed (see [Pu4]).

In [Co1] A.Corti showed that methods of [IsMa] solved a much more general problem

then �nding all birational automorphisms of a given variety. For example, Quartic Theorem

could be generalized in the following way.

Generalized Quartic Theorem. Smooth quartic 3-fold in P4 is not birational to Mori

3-folds2, which are not isomorphic to X; conic bundles; �brations on surfaces with Kodaira

dimension �1.

Recently A.V.Pukhlikov found the local analog of the methods of V.A.Iskovskikh and

Yu.I.Manin in paper [Pu4]. This allowed him to give a very elegant proof of the following

generalization of Quartic Theorem in paper [Pu5].

Fano Hypersurfaces. Every birational automorphism of general enough hypersurface

X � PN of degree N � 5 is the identity map.

Using local methods of [Pu4] A.V.Pukhlikov solved the following old and classical

problem on the birational rigidity of �brations on del Pezzo surfaces (see [Pu7]).

Del Pezzo Fibrations. Let � : X ! S be \very rami�ed"3 del Pezzo �bration, such

2Mori 3-folds are Fano 3-folds with terminal Q-factorial singularities and Picard group Z.3K2

X62 NE (X), where NE (X) is a closure of a cone of e�ective 1-cycles on X.

3

that X is smooth, Pic(X=S) �= Z and

X��=

8>>><>>>:double cover of quadric cone in P3K(S) rami�ed in quartic;

double cover of P2K(S) rami�ed in smooth quartic curve;

cubic in P3K(S):

Then the following exact sequence of groups exists

1! Bir(X�)! Bir(X)! Bir(S):

Last year A.Corti found in [Co2] the link between questions of birational rigidity

and such modern and powerful tool of Log Minimal Model Program as V.V.Shokurov's

Connectedness Theorem (see [Sh2]). The latter helped him to obtain new simple proofs of

classical results on birational rigidity.

I expect that appications of V.V.Shokurov's Connectedness Theorem in birational

geometry are much more notable than they are understood to be now. So we should

expect new achievements in birational geometry very soon.

4

x2. Obtained results.

A couple of years ago, inspired by the Generalized Quartic Theorem, I asked myself the

following natural question.

Question. In which Fano 3-folds with canonical singularities, elliptic �brations and

�brations on surfaces with Kodaira dimension zero can there be a birationally transformed

smooth quartic 3-fold in P4?

Note that a quartic 3-fold in P4 can be birationally transformed into an elliptic �bration!

Construction of Elliptic Fibration. Choose a smooth quartic 3-fold X in P4 and

line L � X. Consider projection : X 9 9 KP2 from L and resolution of the indeterminacy

of rational map W

f . & g

X 9 9 K P2

:

Then g is an elliptic �bration.

Smooth quartic 3-fold in P4 contains a line. Therefore, the Question could not have a

negative-type answer similar to the statement of the Generalized Quartic Theorem.

Similar to the Construction of Elliptic Fibration one can birationally transform smooth

quartic 3-fold in P4 into �brations on K3 surfaces.

Construction of K3 Fibration. Take any smooth quartic X in P4 and a pencil P of

its hyperplane sections. Consider the resolution of indeterminacy of map �P

W

f . & g

X�P9 9 K P1

:

Then g is a �bration on K3 surfaces.

I explicitly answered the Question in paper [Ch1] in the following way.

Answer. Smooth quartic 3-fold in P4 is not birationally isomorphic to Fano 3-folds

with canonical singularities, which are not biregular to X; elliptic �brations and �brations

on surfaces with Kodaira dimension zero except �brations described above in Constructions

of Elliptic and K3 Fibrations4.

4I will identify �brations, that are birationally equivalent as �brations.

5

I developed a method that could solve the following problem for most birationally rigid

varieties.

Problem. Describe birational maps of a given variety X into such �bration

� : V ! Z;

that the general �ber of � has Kodaira dimension zero.

I explicitly solved Problem for the following smooth varieties: the double cover of P3

rami�ed in sextic, quartic 3-fold in P4, double cover of quadric in P3 rami�ed in octic,

\very rami�ed" conic bundle, general hypersurface of degree M in PM .

Note that as a corollary of my results a double cover of P3 rami�ed in smooth sextic and

\very rami�ed" 3-fold conic bundle could not be birationally transformed into any elliptic

�bration. Nowadays these are the only examples of rationally connected varities that could

not be birationally represented as an elliptic �bration.

Most of the known birationally rigid varieties are smooth. However, some singular 3-

folds are known to be birationally rigid. Among them there are a double cover of P3

rami�ed in sextic with one simple double point and a quartic 3-fold in P4 with one simple

double point. I solved the Problem for both of them.

In particular, I proved that every double cover of P3 rami�ed in sextic with one simple

double point could be birationally transformed into a unique elliptic �bration. Thus one

can consider this variety as a birationally rigid elliptic �bration! As far as I know there are

no other rationally connected varieties, that are known to have such property.

Applying my method in the two-dimensional case I solved the Problem for smooth del

Pezzo surfaces of small degree and \very rami�ed" surface conic bundles over an arbitrary

algebraically non-closed �eld.

I proved birational rigidity and solved the Problem for an arbitrary smooth hypersurface

of degree N in PN in cases N = 5; 6; 7 or 8. In particular, I obtained the following result.

Fano Hypersurfaces of Small Degree. Let X be an arbitrary smooth hypersurface

X � PN of degree N = 5; 6; 7 or 8. Then Bir(X) = Aut(X).

My method of solving the Problem could be applied to classi�cation of all birational

transformations of birationally rigid varieties into Fano �brations with canonical singula-

rities. The latter coincides with the regular birational rigidity analysis when the base of

6

such Fano �bration is a point, thanks to Minimal Model Program. Nevertheless, birational

maps between a given birationally rigid variety and an arbitrary Fano variety with canonical

singularities are a rather new topic and heve never been studied before.

I proved that all varieties mentioned above except a double cover of quadric and singular

quartic can not be birationally transformed into any Fano 3-fold with canonical singulari-

ties. Moreover, I classi�ed all Fano 3-folds with canonical singularities that are birationally

isomorphic to the smooth double cover of the quadric 3-fold (one-dimensional family) and

singluar quartic 3-fold (exactly 25 Fano 3-folds).

I revealed one interesting phenomenon. In all studied cases Fano 3-folds with canonical

singularities, that are birationally equivalent to a given 3-fold, are in one-to-one correspon-

dence with generators of the group of birational automorphisms of this given 3-fold. Then

I explicitly checked that a similar result holds for del Pezzo surfaces over algebraically

non-closed �elds as well. Finally, I showed that such coincidence follows from the general

rule of regularization of biregular automorphisms.

7

x3. Thesis' description.

All varieties in the thesis are assumed to be projective and de�ned over C unless the

opposite is explicitly speci�ed. Main de�nitions, notations and notions are contained in

paper [KaMaMa]. All results of the thesis are contained in [Ch1-7].

Chapter I is dedicated to the properties of rather new objects { movable log pairs.

Their global and local properties are studied by means of Minimal Model Program and

applications of V.V.Shokurov's Connectedness Theorem (see [Sh2]).

In Chapter II old and new notions of birational rigidity are discussed. This chapter

contains some necessary conditions and criteria on a variety to be birationally rigid.

The entire Chapter III is dedicated to three smooth del Pezzo surfaces with Picard

group Z de�ned over an algebraically non-closed �eld. The importance of this chapter is

due to clear geometric constructions and the simplest way to show application of results

of Chapter II.

Three-dimensional analog of Chapter III is Chapter IV. This chapter deals with three

smooth Fano 3-folds: a double cover of P3 rami�ed in smooth sextic, a smooth quartic

3-fold in P4 and a double cover of quadric 3-fold rami�ed in smooth octic.

Conic bundles are considered in Chapter V.

Chapter VI is about birational rigidity of arbitrary smooth hypersurfaces of degree N

in PN in cases N = 5; 6; 7 or 8. The main result of this chapter is a special case of the

general conjecture of V.A.Iskovskikh.

Two singular 3-folds are studied in Chapter VII. This chapter contains an interesting

example of one elliptic �bration, that has properties similar to birational rigidity of a conic

bundle.

In Appendix I study questions of regularization of birational automorphisms. An

algorithm for explicit regularization is given there and then applied to birational involutions

on del Pezzo surfaces and a double cover of a quadric 3-fold rami�ed in smooth octic.

Appendix also contains a description of all �nite subgroups of the group of birational

automorphisms of some algebraic varieties.

8

LOG PAIRS ON ALGEBRAIC VARIETIES

x1. Movable log pairs.

This section introduces objects, that will be permanently used in the thesis.

Main Object. Movable log pair

(X;MX) = (X;nXi=1

biMi)

is a variety X together with a formal �nite linear combination of linear systemsMi without

�xed components, such that all bi 2 Q�0 .

Note, that (X;MX) can be considered as a usual log pair (see [KaMaMa]). Indeed, one

can replace every linear system Mi by its general members or by appropriate weighted

sum. In particular, one can use log canonical divisor KX +MX and boundary MX as

usual divisors.

Observation. Strict transform of MX is naturally de�ned for any birational map.

I will assume that log canonical divisors of all considered log pairs are Q -Cartier divisors.

So, discrepancies, terminality, canonicity, log terminality and log canonicity can be de�ned

for movable log pairs in the similar way as for usual ones.

Application of Log Minimal Model Program to canonical (terminal) movable log pair

preserves its canonicity (terminality).

Center of Canonical Singularities. Proper irreducible subvariety Y � X is a center

of canonical singularities of (X;MX), if there is a birational morphism f : W ! X and

f -exceptional divisor E � W , such that a(X;MX ; E) � 0 and f(E) = Y .

Set of Centers of Canonical Singularities. CS(X;MX) will denote a set of all

centers of canonical singularities of (X;MX).

The following example will clarify the nature of just introduced objects.

Simple Example. Take movable log pair (P2; bM), where M is a linear system of lines

passing through point O. Then

CS(P2; bM) =

(; for b < 1;

fOg for b � 1:

9

This example leads to the following notice.

Observation. Singularities of movable log pair coincide with singularities of variety

outside of the base locuses of components of the boundary.

Later the following type of singularities will be rather useful.

Semiterminal Singularities. Movable log pair (X;MX) has semiterminal singulari-

ties if for some rational � > 1 movable log pair (X;�MX) has canonical singularities.

The following example sheds light on the nature of semiterminal singularities.

Standart Example. Consider quadric cone Q in P3 and complete linear systemM of

hyperplane sections of Q. Then log pair (Q; bM) is semiterminal for all b 2 Q�0 .

It is natural to ask, is there some special model of a given log pair?

Canonical Model. I will say, that (V;MV ) is a canonical model of (X;MX), if there

is a birational map : X 9 9 KV , such that

(V;MV ) = (V; (MX));

divisor KV +MV is ample and (V;MV ) has canonical singularities.

The following important theorem holds.

Uniqueness Theorem. Canonical model is unique if it exists.

To prove Uniqueness Theorem one just need to switch canonical divisor with log

canonical one in the proof of uniqueness of canonical model of algebraic variety.

General Type. I will say that movable log pair (X;MX) is of general type if canonical

model of (X;MX) exists.

Let us consider the following example.

V.A.Iskovskikh's Example. Consider smooth Fano 3-fold V with very ample anti-

canonical divisor, such than �K3V = 16 and Pic(V ) = Z (see [Is2]). Let HC be a linear

system of hyperplane sections of V passing doubly through a general enough line C � V

and b 2 Q>4 . Then canonical model of movable log pair (V; bHC) is (P3; bjOP3(1)j).

For an arbitrary movable log pair (X;MX) consider birational morphism f : W ! X,

such that log pair

(W;MW ) = (W; f�1(MX))

10

has canonical singularities.

Iitaka Map and Kodaira Dimension. If jn(KW +MW )j 6= ; for n� 0, then map

I(X;MX) = �jn(KW+MW )j Æ f�1 for n� 0

is called Iitaka map of (X;MX) and

�(X;MX) = dim(I(X;MX)(X))

is called Kodaira dimension of (X;MX). Otherwise I(X;MX) is considered to be unde�ned

everywhere and �(X;MX) = �1.

One can prove the following statement.

Correctness Theorem. Map I(X;MX) and number �(X;MX) do not depend on the

choice of morphism f .

Note, that Iitaka map and Kodaira dimension of movable log pair apriori depend on

natural number n� 0 in their de�nition.

Remark. Kodaira dimension of movable log pair depends only on the properties of

movable log pair.

Consider the following example.

Example. Let X be double cover of P3 rami�ed in smooth sextic and

(X;MX) = (X;NXi=1

biMi)

be some movable log pair. Fix such rational number � 2 Q>0 [ f+1g, that

KX + �MX �Q 0

and � = +1 in case MX = ;. Then

�(X;BX) =

8>>>>>><>>>>>>:

�1 for � > 1;

0 for � = 1;

1 for � < 1 and allMi are compounded of one pencil P � j �KX j;

3 otherwise.

11

Note, that Log Abundance Conjecture implies that Iitaka map is independent on the

choice of number n � 0. Unfortunately, Log Abundance is proved only for surfaces and

3-folds (see [KeMaMe]).

Remark. Posteriori Iitaka maps of all movable log pairs in the thesis do not depend on

number n� 0 in the last de�nition.

If �(X;MX) 6= dim(X) canonical model does not exist. Therefore, sometimes it is

convenient to use the following model.

Weakly Canonical Model. Movable log pair (V;MV ) is a weakly canonical model of

movable log pair (X;MX), if there is a birational map : X 9 9 KV , such that

(V;MV ) = (V; (MX));

divisor KV +MV is nef and (V;MV ) has canonical singularities.

Weakly canonical model may not exist and may not be unique.

Note. I will use mostly movable log pairs. So, sometimes I will call them simply log

pairs. I hope it will make no confusion.

12

x2. Global properties of log pairs.

In this section I will consider general properties of log pairs on Mori �brations.

Fix Mori �bration � : X ! S together with log pair

(X;MX) = (X;nXi=1

biMi)

with �(X;MX) 2 [0; dim(X)) and � 2 Q \ (0; 1], such that

KX + �MX �Q ��(L)

for Q -Cartier divisor L on S.

The following result is an analog of inequality of M.N�other and G.Fano.

M.N�other & G.Fano & V.A.Iskovskikh's Theorem. Suppose S is a point and

log pair (X;�MX) is terminal. Then � = 1, �(X;MX) = 0 and (X;MX) has no weakly

canonical models except itself.

Proof. Suppose � < 1. Consider Æ 2 Q \(�; 1), such that log pair (X; ÆMX) is terminal.

Then

dim(X) = �(X; ÆMX) � �(X;MX) < dim(X):

So, � = 1, (X;MX) is terminal and �(X;MX) = 0.

Assume the existence of the following commutative diagram

W

f . & g

X�9 9 K Y

;

such that variety W is smooth, morphisms f and g are birational, log pair

(Y;MY ) = (Y; �(MX))

is canonical and KY +MY is nef. Then

kXj=1

a(X;MX ; Fj)Fj �Q g�(KY +MY ) +

lXi=1

a(Y;MY ; Gi)Gi;

13

where divisors Gi and Fj are exceptional for morphisms g and f correspondingly. From

Lemma 2.19 of paper [Ko1] follows, that for all divisors E on variety W

a(X;MX; E) = a(Y;MY ; E):

In particular,

KY +MY �Q 0;

log pair (Y;MY ) is terminal and k = l.

Pic(X) = Z and Q -factoriality of variety X imply

rk(Pic(W )) = 1 + k:

On the other hand,

rk(Pic(W )) � rk(Pic(Y )) + l:

Hence variety Y is Q -factorial and Pic(Y ) = Z.

Now I can consider � 2 Q>1 , such that both (X; �MX) and (Y; �MY ) are canonical

models. Uniqueness Theorem implies � is an isomorphism. �

Conditions of M.N�other & G.Fano & V.A.Iskovskikh's Theorem could not be weakened.

Example. Consider smooth quartic 3-fold V and linear system HC of hyperplane

sections of V passing through line C � V . Put MV = bHC for some b 2 Q>1 . Then

KV +1

bMV �Q 0;

singularities of log pair (X; 1bMV ) are canonical and �(V;MV ) = 1.

I need an analog of M.N�other & G.Fano & V.A.Iskovskikh's Theorem in the case when

variety S is not a point.

V.G.Sarkisov's Theorem. Assume (X;�MX) is canonical and L is nef and big. Then

there is a dominant map

: I(X;MX)(X) 9 9 KS;

such that � = Æ I(X;MX).

Proof. Let � = 1. Then I(X;MX) = jn��(L)j for n � 0. Last implies existence of

birational map .

14

Let now � 2 (0; 1). Consider birational morphism f :W ! X, such that log pair

(W;MW ) = (W; f�1(MX))

is terminal and W is smooth. Then

KW +MW �Q f�(��(L)) +

kXi=1

a(X;�MX ; Fi)Fi + (1� �)MW ;

where divisors Fi are f -exceptional and all a(X;�MX ; Fi) � 0. By de�nition

I(X;MX) = I(W;MW ) Æ f�1:

Last implies existence of map . �

Corollary of V.G.Sarkisov's Theorem. Suppose that in V.G.Sarkisov's Theorem

morphism � and map I(X;MX) are not birationally equivalent. Then (X;�MX) is not

terminal in the neighborhood of the general �ber of �.

During the proof of V.G.Sarkisov's Theorem one can notice the following.

Observation. In conditions of V.G.Sarkisov's Theorem the following statements are

equivalent: is birational; � = 1; �bers of I(X;MX) are uniruled; �(X;MX) = dim(S).

Let us give without a proof one generalization of V.G.Sarkisov's Theorem.

Generalization of V.G.Sarkisov's Theorem. Let (X;�MX) be canonical log pair.

Then there is a dominant map

: I(X;MX)(X) 9 9 K�jnLj(S);

such that Æ I(X;MX) = �jnLj Æ � for n� 0.

15

x3. Local properties of log pairs.

In this section I will consider applications of V.V.Shokurov's Vanishing Theorem.

Fix usual log pair

(X;BX) = (X;kXi=1

biBi);

where all bi 2 Q \ [0; 1] and divisor KX +BX is Q -Cartier.

Center of Log Canonical Singularities. Proper irreducible subvariety Y � X is

a center of log canonical singularities of log pair (X;BX), if there are such birational

morphism f :W ! X and divisor E �W , that

a(X;BX ; E) � �1 and f(E) = Y:

Note, that center of log canonical singularities has local nature. Nevertheless, one can

consider its global analog.

Set of Centers of Log Canonical Singularities. LCS(X;BX) will denote a set of

all centers of log canonical singularities of log pair (X;BX).

Now I will describe scheme structure on set LCS(X;BX).

Fix some birational morphism f : W ! X, such that W is smooth and union of all

strict transforms of divisors Bi on W and f -exceptional divisors forms divisor with simple

normal crossing.

Consider such log pair (W;BW ), that

KW + BW �Q f�(KX + BX)

and f(BW ) = BX .

Morphism f is usually called log resolution of (X;BX) and log pair (W;BW ) is usually

called log pull back of log pair (X;BX)

Log Canonical Singularities Subscheme. Subscheme associated with ideal sheaf

I(X;BX) = f�(d�BY e)

is called log canonical singularities subscheme of log pair (X;BX) and denoted as L(X;BX).

16

Note, that every center of log canonical singularities of log pair (X;BX) is contained in

the support of scheme L(X;BX).

Locus of Log Canonical Singularities. Support of subscheme L(X;BX) is called

locus of log canonical singularities of log pair (X;BX) and denoted as LCS(X;BX).

Note, that there is a little ambiguity in considering LCS(X;BX) to be a subvariety and

set simultaniously.

V.V.Shokurov's Vanishing Theorem. Suppose for some nef and big divisor H on

variety X divisor KX + BX +H is Cartier. Then for all i > 0

Hi(X; I(X;BX)D) = 0:

Proof. By Kawamata-Viehweg vanishing for all i > 0

Rif�(f�(KX + BX +H) + d�BW e) = 0:

Degeneration of local-to-global spectral sequence and

R0f�(f�(KX +BX +H) + d�BW e) = I(X;BX)D

imply that for all i

Hi(X; I(X;BX)D) = Hi(W; f�(KX +BX +H) + d�BW e):

On the other hand, for all i > 0

Hi(W; f�(KX + BX +H) + d�BW e) = 0

by Kawamata-Viehweg vanishing. �

For any Cartier divisor D on X there is an exact sequence

0! I(X;BX)D ! OX(D)! OL(X;BX )(D)! 0:

Therefore, V.V.Shokurov's Vanishing Theorem implies the following result.

17

V.V.Shokurov's Global Connectedness Theorem. Suppose �(KX + BX) is nef

and big. Then LCS(X;BX) is connected.

Consider another application of V.V.Shokurov's Vanishing Theorem.

Quadric Lemma. Let (V;BV ) be log pair, such that V �= P1 � P1 and divisor BV has

type (a; b) in Pic(V ) for some a and b in Q \ [0; 1). Then LCS(V;BV ) = ;.

Proof. Intersecting boundary BV with the rulings of V I concludes that there are no

curves on V , which belong to LCS(V;BV ). So, I may assume that LCS(V;BV ) contains

some point O 2 V .

Let H be divisor on V , which has type (1�a; 1�b) in Pic(V ). Then H ample. Moreover,

divisor

D = KV + BV +H

is Cartier and H0(OV (D)) = 0.

On the other hand, V.V.Shokurov's Vanishing Theorem implies surjectivity

H0(OV (D))! H0(OL(V;BV )(D))! 0:

�

Now I will state relative version of V.V.Shokurov's Global Connectedness Theorem.

V.V.Shokurov's Relative Connectedness Theorem. Suppose that for some

morphism g : X ! Z divisor �(KX + BX) is g-nef and g-big. Then LCS(X;BX) is

connected in the neighbourhood of any �ber of g.

I de�ned previously center of canonical singularities only for movable log pairs. Neverthe-

less, in the given de�nition I never used movability of log pair. Now I will apply V.V.Sho-

kurov's Relative Connectedness Theorem to trace one link between centers of canonical

singularities and centers of log canonical singularities.

Inverse of Adjunction. Let Z 2 CS(X;BX) and H be e�ective irreducible and smooth

Cartier divisor on X, such that H contains Z and H is not a component of boundary BX .

Then log pair LCS(H;BX jH) 6= ;.

Proof. Let f :W ! X be log resolution of (X;BX +H) and H = f�1(H). Then

KW + H �Q f�(KX +BX +H) +

XE 6=H

a(X;BX +H;E)E:

18

Note, that

fZ;Hg � LCS(X;BX +H):

Therefore, application of V.V.Shokurov's Relative Connectedness Theorem to log pulback

of log pair (X;BX +H) on W leads to

H \ E 6= ;

for some divisor E 6= H on W , such that a(X;BX ; E) � �1. Now claim follows from

KH � (KW + H)jH �Q f j�H(KH + BX jH) +

XE 6=H

a(X;BX +H;E)EjH:

�

Now I will show on application of Inverse of Adjunction and Quadric Lemma. Namely,

I will prove the following result, which is Theorem 3.11 of paper [Co].

A.Corti's Theorem. Suppose that O 2 CS(X;BX), where O is a simple double point

of 3-fold X. Then multO(BX) � 1:

Proof. Let f :W ! X be a blow up of point O, Q = f�1(O) and

(W;BW ) = (W; f�1(BX)):

Then

KW +BW �Q f�(KX + BX) + (1�multO(BX))Q:

Therefore, divisor Q contains some element Z of set CS(W;BW ). So, from Inverse

of Adjunction follows that singularities of log pair (Q;BW jQ) are not log terminal. Last

contradicts Quadric Lemma. �

Next application of Inverse of Adjunction will use the folowing slightly modi�ed version

of Theorem 3.1 of [Co].

A.Corti's Lemma. Let O be a smooth point of surface H and for some non-negative

rational numbers a1 and a2

O 2 LCS(H; (1� a1)�1 + (1� a2)�2 +MH);

19

where boundary MH is movable and irreducible reduced curves �1 and �2 intersect

normally in point O. Then

multO(M2H) �

(4a1a2 in case a1 � 1 or a2 � 1;

4(a1 + a2 � 1) in case a1 > 1 and a2 > 1:

Note, that most applications use only special case of A.Corti's Lemma.

Special Case of A.Corti's Lemma. Suppose for smooth point O on surface H and

movable log pair (H;MH) set LCS(H;MH) contains point O. Then

multO(M2H) � 4

and equality implies mult(MH) = 2.

Now I will show how to prove classical result of V.A.Iskovskikh in the modern form of

A.V.Pukhlikov using method of A.Corti.

V.A.Iskovskikh & A.V.Pukhlikov's Theorem. Let O be such a smooth point on

3-fold X, (X;MX) be a movable log pair, such that set CS(X;MX) contains point O. Then

the following inequality holds

multO(M2X) � 4

and equality implies multO(MX) = 2.

Proof. Let H be a general enough very ample divisor on X passing through point O.

Then

multO(MX) = multO(MX jH)

and

multO(M2X) = multO((MX jH)

2):

Inverse of Adjunction implies

O 2 LCS(H;MX jH):

Now claim follows from Special Case of A.Corti's Lemma. �

20

BIRATIONAL RIGIDITY

x1. What is birational rigidity?

In this section I will try to explain the evolution of the notion of birationally rigid variety

since its appearance and untill nowadays.

So, what is birational rigidity? As far as I know notion of birationally rigid variety

appeared �rst in early papers of V.A.Iskovskikh and Yu.I.Manin. However implicitly it

have been used by M.N�other and G.Fano long time ago.

Emergence of the birational rigidity was closely related with the intensive study of the

group of birational automorphisms of surfaces and 3-folds during this century. So, �rst

de�nition of birational rigidity was the following.

Ancient Birational Rigidity. Variety X is birationally rigid if Bir(X) = Aut(X).

In particular, Quartic Theorem (see Introduction) simply means that smooth quartic

3-folds are birationally rigid in the given de�nition.

Rational higher-dimensional varieties have huge group of birational automorphisms.

Therefore, words \birationally rigid" were used as an antonym to the word \rational"

for a long time.

Note, that to prove birational rigidity is one of few ways to prove non-rationality of

higher-dimensional rationally connected variety. Moreover, for some varieies to prove their

birational rigidity is the only possible way of proving their non-rationality!

Example. Nowadays non-rationality of a given smooth quartic (not just general one)

could be proved only via its birational rigidity.

Just after the quartic 3-fold V.A.Iskovskikh found generators of the groups of birational

automorphisms of the double cover of smooth quadric 3-fold rami�ed in smooth octic

and smooth intersection of quadric and cubic in P5. From the old point of view both

these 3-folds were not birationally rigid, while most of their properties resemble properties

of smooth quartic 3-fold a lot. Thus, notion of birational rigidity transformed into the

following (for details see [Is3]).

21

Original Birational Rigidity. Variety X is birationally rigid if \maximal singulrities"

of every linear system on X without �xed components could be \untwisted" by means of

given birational automorphisms.

Nevertheless, smooth quartic 3-fold �ts much more narrow class of varieties of the

following kind.

Birational Superrigidity. Variety X is called birationally superrigid if every linear

system on X without �xed componends has no \maximal singularities".

Using classi�cation of all smooth Fano 3-folds V.A.Iskovskikh managed to show that

his technique gives much more then description of the group of birational automorphisms.

In particular, he proved that all considered birationally rigid smooth Fano 3-folds are not

birationally isomorphic to smooth Fano 3-folds non-biregular to them.

Remark. Examples of birationally rigid varieties showed that birational rigidity is a

very rare property among varieties with negative Kodaira dimension.

Everything was �ne untill the results of V.G.Sarkisov on higher-dimensional conic

bundles. His conic bundles behaved very similar to the birationally rigid varieties, but

didn't satis�ed any de�nition of birational rigidity, which used the properties of linear

systems. Last leaded to the following de�nition of birationally rigid conic bundle, which

perfectly correlates with the ancient de�nition of birational rigidity.

Birationally Rigid Conic Bundle. Conic bundle � : X ! S is called birationally

rigid once the following exact sequence of groups exists

1! Aut(X�)! Bir(X)! Bir(S);

where X� is geometric generic �ber of morphism �.

Later A.V.Pukhlikov considered �bration on del Pezzo surfaces in the framework of

birational rigidity and naturally extended de�nition of birationally rigid conic bundle to

all kinds of �brations.

Birationally Rigid Fibration. Fibration � : X ! S is called birationally rigid once

geometric generic �ber X� of morphism � is birationally rigid and the following exact

sequence of groups exists

1! Bir(X�)! Bir(X)! Bir(S):

22

For a very long time \very rami�ed" conic bundles were the only known examples of

birationally rigid �brations. Nevertheless, work of V.G.Sarkisov and A.V.Pukhlikov showed

that probably all \very rami�ed" �brations are birationally rigid. The only problem here

is to determine what is \very rami�ed" means.

Remark. In some sense \most" of all 3-folds with negative Kodaira dimension are

birationally rigid.

Minimal Model Program sheded new light on the problems of birational rigidity by

means of the article [Co1] of A.Corti. After his article it became clear that one should

better consider birational rigidity only for so-called Mori �brations, i.e. �brations with

relatively ample anticanonical divisor and terminal Q -factorial singularities, which relative

Picard group is Z. For such �brations one can consider birational rigidity in the following

form.

Modern Birational Rigidity. A given Mori �bration � : X ! S is called birationally

rigid (superrigid) if for an arbitrary Mori �bration � : Y ! Z existence of birational map

� : X 9 9 K Y induces the isomorphism of the geometric generic �bers X� and X� and

existence of commutative diagram

X�9 9 K Y

� # # �

S 9 9 K Z

;

where map is birational as well (map is birational and birational map � is an

isomorphism of general �bers of �brations � and �).

Note, that output of Minimal Model Program is either Mori �bration or minimal model

with the structure of Iitaka �bration, which is almost uniquely de�ned from birationally

point of view. Therefore, Mori �brations were always considered as a relatively minimal

model, which is not uniquely de�ned in the birational class of a given variety.

Observation. Birationally superrigid varieties are unique Mori �brations in its birati-

onal class.

So, birationally rigid �brations looks like almost uniquely de�ned relatively minimal

models of varieties with negative Kodaira dimesnion. Moreover, birationally superrigid

23

�brations plays the role similar to Iitaka �brations of varieties with positive Kodaira

dimension. In particular, birationally superrigid varieties plays the role similar to the

role of canonical models for the varieties of general type.

Remark. Modern de�nition of birational rigidity generalizes all previous de�nitions of

birational rigidity.

So, �nally it looks like notion of birationally rigid �bration and variety is settled.

However, appearance of paper [Gr], careful rereading of paper [Kh] and anouncement of

recent result of A.Corti and M.Mella (paper is not �nished yet) weaken the basement of

the last given de�nition of birational regidity.

As I already explained words \birationally rigid" stands for some kind of uniqueness.

Last is easy to admit, because of similarity to some uniqueness results for varieties with

positive Kodaira dimension. Nevertheless, M.Grinenko in paper [Gr] gave an example

of singular 3-fold, which can be represented as a Mori �bration in two di�erent ways

up to birationally equivalence of �brations. Moreover, many years ago S.I.Khashin in

underestimated paper [Kh] (in my opinion almost all proofs of paper [Kh] are incorrent,

while all statements are true) revealed that one rather famous 3-fold can be represented as

a Mori 3-fold in a unique way and as Mori �brations in in�nitely many ways (actually all

such representations can be parametrized by points on P2).

Rhetorical Question. So, what is birational rigidity?

My point of view is the following.

Opinion. Variety X may be considered birationally rigid, if one can \e�ectively

describe" all Mori �brations birationally equivalent to X.

Thus, all surfaces with negative Kodaira dimension are birationally rigid in the last

sense. Moreover, going further one can say that theoretically all varieties with negative

Kodaira dimension are birationally rigid.

Remark. In the rest of the thesis we will use Modern Birational Rigidity only.

24

x2. Conditions and criteria.

In this section I will give some conditions and criteria for being birationally rigidit in

modern sense.

Fano Variety. Let V be an arbitrary Fano variety with Q -factorial terminal singulari-

ties and Pic(V ) = Z.

When V is supperrigid? In the sense, that V is unique up to isomorphism Mori �bration

in the class of all varieties birationally equivalent to V .

Condition on Superrigidity. Variety V is birationally superrigid if every movable log

pair (V;MV ) with KV +MV �Q 0 has canonical singularities.

Proof. Suppose that � is a birational transformation of V into such �bration � : Y ! Z,

that general �ber of � has Kodaira dimension �1. I will prove that � is an isomorphism.

Assume that Z is not a point and take \big enough" very ample divisorH on Z. Consider

such � 2 Q>0 and movable log pair

(V;MV ) = (V; ���1(j��(H)j));

that KV +MV �Q 0. Due to condition of the theorem log pair (V;MV ) has canonical

singularities. Thus, �(V;MV ) = 0. On the other hand, �(V;MV ) = �1 by construction.

Therefore, Z is a point.

Consider integer n� 0, positive rational number � and movable log pairs

(Y;MY ) = (Y;�

nj � nKY j) and (V;MV ) = (V; ��1(MY );

such that KV +MV �Q 0. Then canonicity of singularities of log pair (V;MV ) implies

�(V;MV ) = �(Y;MY ) = 0:

Therefore � = 1, which contradicts M.N�other & G.Fano & V.A.Iskovskikh Theorem (see

section 2 of Chapter I). �

Actually, one can show that Condition on Superrigidity is a criterion up to existence of

Log Minimal Model Program.

25

Criterion of Superrigidity. Suppose that V is three-dimensional. Then V is

birationally superrigid if and only if every movable log pair (V;MV ) with KV +MV �Q 0

has canonical singularities.

Proof. Due to Condition on Superrigidity I have to prove the \only if" part. Suppose

the existence of such movable log pair (V;MV ), that

KV +MV �Q 0

and singularities of (V;MV ) are worse than canonical. Therefore, �(V;MV ) = �1.

So, application of Log Minimal Model Program to log pair (V;MV ) gives non-trivial

birational transformation of variety V into some Mori �bration. Thus, V is not birationally

superrigid. �

To proceed to other conditions on birational rigidity I have to introduce some kind of

minimality of a movable log pair.

Minimal Log Pair. Movable log pair (V;MV ) is minimal if for any � 2 Bir(V ) divisor

�(MV )�MV

is nef.

Importance of minimality of movable log pair is due to the following.

Easy Statement. Every movable log pair on Fano variety V is birationally equivalent

to a minimal one.

Proof. Let H be an ample generator of Picard group of V and ind(V ) be global

Gorenstein index of V . Fix an arbitrary movable log pair

(V;MV ) = (V;nXi=1

biMi):

Then every birational map g 2 Bir(V ) induces positive rational number

�(g) =nXi=1

bidi(g); where g(Mi) �Q di(g)H and di(g) 21

ind(V )N :

Construction of numbers �(g) implies that they satisfy descending chain condition. In

particular, they have minimal element �max, which corresponds to some not uniquely

de�ned birational map gmax 2 Bir(V ). Put

(V;MmaxV ) = (V; g(MV )):

26

Then for any 2 Bir(V ) divisor

(MmaxV )�Mmax

V = (�( Æ gmax)� �max)H

is nef. �

The following simple property of minimal log pairs is rather convienient some times.

Observation. If movable log pair (V;MV ) is minimal, then log pair (V; �MV ) is also

minimal for any � 2 Q>0 .

Described properties of minimal log pair together with Condition on Superregidity imply

the following.

Condition on Rigidity. Variety V is birationally rigid if every minimal movable log

pair (V;MV ) with KV +MV �Q 0 has canonical singularities.

Moreover, as in supperrigidity case existence of Log Minimal Model Program implies

that that Condition on Rigidity is criterion.

Criterion on Rigidity. Suppose that V is three-dimensional. Then V is birationally

rigid if and only if every minimal movable log pair (V;MV ) with KV + MV �Q 0 has

canonical singularities.

Proof. Suppose that V is birationally rigid and some movable log pair (V;MV ) is

minimal,

KV +MV �Q 0

and singularities of (V;MV ) are worse then canonical. Then �(V;MV ) = �1.

Application of Log Minimal Model Program to movable log pair (V;MV ) gives non-

trivial birational transformation of variety V into some Mori �bration � : Y ! Z, such

that log pair

(Y;MY ) = (Y; �(MV )

has terminal singularities and divisor �(KY +MY ) is � -ample.

On the other hand, birational rigidity of V implies that Z is a point and Y �= V . Thus,

divisor �(MV )�MV is not nef, which contradicts to minimality of (V;MV ). �

Now I will consider similar topics for �brations.

Mori Fibration. Let � : X ! S be a Mori �bration.

27

How can above conditions and criteria be generalized for Mori �bration �?

Bad Condition on Fibration Superrigidity. Fibration � : X ! S is birationally

superrigid if every movable log pair (X;MX) on X with Q -rationally �-trivial log canonical

divisor has canonical singularities and nef log canonical divisor KV +MV .

Proof. Proof is a corollary of Theorem 4.2 of [Co1]. �

Just given condition is bad, because it looks like except the case S is a point there are

no uniruled �brations satisfying this condition. Indeed, one can check that conditions of

the Bad Condition on Fibration Rigidity imply that � : X ! S is unique Mori �brations

among all varieties, which are birationally equivalent to X.

Remark. In case when dimension of S is greater than one �bration � could not be the

only Mori �bration, which is birationally equivalent to variety X.

Moreover, it seems that even in the case S is a curve Mori �bration � always has has

di�erent birational representation as a Mori �bration. At least there are no examples of

Mori �bration over a curve, which could not be non-trivially birationally transformed into

Mori �bration preserving the general �ber.

Observation. In all given above conditions and criteria on rigidity and superrigidity

of Fano 3-fold V one can use condition �(V;MV ) = 0 or condition �(V;MV ) � 0 instead

of condition that singularities of movable log pair (V;MV ) are canonical.

Proof. Indeed, for movable log pair (X;MX) with KX + MX �Q 0 canonicity of

singularities is equivalent to equality �(X;MX) = 0 or inequality �(V;MV ) � 0. �

Thus, it is rather naturally to expect that to get proper generalization of Condition on

Superrigidity for �bration � one probably should exchange words \canonical singularities

and nef log canonical divisor" by words \Kodaira dimension is non-negative" in Bad

Condition on Fibration Superrigidity. However, I could not prove such kind of condition

without using Log Minimal Model Program. Therefore, I will consider only the 3-fold case,

where such condition becomes a criterion.

Criterion on Fibration Superrigidity. Suppose that X is a 3-fold. Then Mori

�bration � : X ! S is birationally superrigid if for every movable log pair (X;MX) on

3-fold X with Q -rationally �-trivial log canonical divisor inequality �(X;MX) � 0 holds.

Proof. At �rts consider \if" part of the claim. Suppose the existence of some Mori

28

�bration � : Y ! Z and birational map � : X 9 9 K Y . I should prove the existence of

commutative diagram

X�9 9 K Y

� # # �

S 9 9 K Z

for some birational map and the fact that � is an isomorphism of general �bers of

�brations � and � .

Note, that case when S is a point is done in Criteria on Superrigidity. Thus, I can

assume that � is either a del Pezzo �bration or a conic bundle.

Fix some \big enough" ample divisor L on Z. Consider integer number n� 0, positive

rational number � and movable log pairs

(Y;MY ) = (Y;�

nj � nKY + ��(L)j) and (X;MX) = (V; ��1(MY );

such that log canonical divisor KX +MX of log pair (X;MX) is Q -rationally �-trivial.

Suppose, that � is a del Pezzo surface. Then application of relative Log Minimal Model

Program to log pair (X;MX) over the curve S gives commutative diagram

X 9 9 K �X

� # # ��

S �= S

;

such that map is birational, �� is Mori �bration and log pair

( �X;M �X) = ( �X; (MX))

has canonical singularities.

Note, that K �X + M �X �Q ���(H) for some Q -Cartier divisor H on curve S. On the

other hand, �( �X;M �X) � 0. Thus, divisor H is nef and Theorem 4.2 of [Co1] implies the

existence of desired commutative diagram and the claim that � is an isomorphism of the

general �bers of �brations � and � .

29

Consider the case when � is a conic bundle. A.Corti in [Co1] proved the existence of

such commutative diagram

X 9 9 K X

� # # �

S� S

;

that map and morphism � are birational, � is Mori �bration and log pair

(X;MX) = (X; (MX))

has canonical singularities.

For some Q -Cartier divisor L on surface S

KX +MX �Q ��(L):

Note, that if divisor L is nef, then we can conclude the proof as in the case of del Pezzo

�bration. So, I can assume that divisor L is not nef.

Paper [KeMaMc] implies the existence of such non-movable boundary BS on surface S,

that BS �Q L and singularities of log pair (S; BS) are log terminal. Thus,

KX +MX �Q ��(KS + BS)

divisor KS + BS is not nef and Log Minimal Model Program implies the existence of

birational morphism q : S ! ~S contracting one irreducible reduced curve C on surface S,

such that

(KS + BS) � C < 0:

Put B ~S = q(BS). Then log pair ( ~S;B ~S) is log terminal and application of relative Log

Minimal Model Program over the surface ~S to movable log pair (X;MX) gives commutative

diagram

Xp9 9 K ~X

� # # ~�

Sq! ~S

;

30

such that map p is birational, ~X has terminal Q -factorial singularities and log canonical

divisor K ~X +M ~X of movable log pair

( ~X;M ~X) = ( ~X; p(MX))

is ~�-nef.

Put G = ��1(C). One can show that rational map p contracts divisor G. Thus, ~� is a

Mori �bration and

K ~X +M ~X �Q ~��(K ~S + B ~S):

Repeating above construction �nitely many times concludes \if" part.

Now I will consider the \only if" part of the criterion. Suppose that there is such

movable log pair (X;MX), that log canonical divisor KX +MX of (X;MX) is Q -rationaly

�-trivial and equality �(X;MX) = �1 holds. Application of Log Minimal Model Program

to log pair (X;MX) gives birational map � of X into such Mori �bration � : Y ! Z, that

log canonical divisor of movable log pair (Y; �(MX)) is � -antiample. Last condition implies

that birational map � could not be an isomorphism of general �bers of � and � . �

Note, that all conditions and criteria considered in this section were explicitly or

implicitly partially used in the papers on birational rigidity. For example, Condition on

Birational Superrigidity became almost the standart part of any such paper. However, the

simple \only if" part of the Criterion on Birational Superrigidity was neglected for a long

time by some obscure reason.

Remark. Notion of \supermaximal singularity" in paper [Pu7] is a phantom of \if"

part of Criterion on Fibration Superrigidity.

One can easily rewrite Criterion on Fibration Superrigidity to handle just birational

rigidity of Mori �bration � : X ! S. Indeed, instead of using minimal log pairs one should

use its relative analog.

Relatively Minimal Log Pair. Movable log pair (X;MX) is relatively minimal if for

any � 2 Bir(X�) divisor �(MX) �MX is �-nef, where X� is geometric generic �ber of

�bration �.

Proof of the following statement is similar to the proof of analogous statement for

minimal log pairs.

31

Statement. Every movable log pair on X is birationally equivalent to a relatively

minimal one.

Combination of the proofs of Criterion on Rigidity and Criterion on Fibration Superri-

gidity leads to the following statement.

Criterion on Fibration Rigidity. Suppose that X is a 3-fold. Then Mori �bration

� : X ! S is birationally rigid if for every minimal movable log pair (X;MX) on 3-fold X

with Q -rationally �-trivial log canonical divisor inequality �(X;MX) � 0 holds.

32

DEL PEZZO SURFACES

x1. Smooth del Pezzo surfaces.

In this section I will consider some well known facts about smooth del Pezzo surfaces of

small degree over an arbitrary �eld F with algebraic closure �F .

Main Object. Let X be a smooth del Pezzo surface over �eld F with

Pic(X) = Z and K2X � 3:

Note, that Pic(X) = Z implies that �eld F is not algebraically closed. Nevertheless, all

results of Chapters I and II still remain valid.

Remark. It is well known (see [Ko2]) that

X �=

8>><>>:hypersurface of degree 6 in P(1; 1; 2; 3) in case K2

X = 1;

hypersurface of degree 4 in P(1; 1; 1; 2) in case K2X = 2;

cubic in P3 in case K2X = 3:

Birational geometry of X was studied by in [Ma1-2], where the following result was

proved.

Del Pezzo Theorem. Bir(X) is generated over Aut(X) by Bertini and Geizer

involutions.

In particular, from Del Pezzo Theorem follows that in case K2X = 1

Bir(X) = Aut(X):

Note, that Del Pezzo Theorem can be complemented in the following way (see [Co1]).

Generalized Del Pezzo Theorem. X is not birationally isomorphic to a conic bundle

and a smooth del Pezzo surface with Picard group Z, which is not biregular to X.

How can surface X be birationally transformed into del Pezzo surfaces with canonical

singularities or into elliptic �brations? In general, the answer depends on the arithmetic

of �eld F. However, something still can be said.

33

Standart Transformation. Suppose that there is a birational morphism f :W ! X,

such that surface W is smooth and K2W > 0. Then for big enough integer number n

birational map �j�nKW j Æ f�1 will be called standart transformation of surface X.

Surface X may have no standart transformations except trivial ones.

Observation. Image of standart transformation of surface X is a del Pezzo surface

with canonical singularities.

Proof. On can see, that morphism f is one of the following: isomorphism; blow up

of F-point O on surface X; blow up of two F-points O1 and O2 on X; consecutive blow

up of two di�erent F-points; blow of a point on X, which splits over the �eld �F in two

components.

Suppose that f is a blow up of F-point O on surface X. Let HO be linear system of

curves in j �KX j passing through the point O. Then

dim(Bs(HO)) = 0;

because divisor �KX is very ample in case K2X = 3 and linear system j �KX j is free and

gives double cover of P2 in case K2X = 2. Thus, equivalence

f�1(HO) � �KW

implies that divisor �KW is nef and big. So, for some integer n� 0 linear system j�nKW j

is free and gives birational map onto normal surface V . Due to construction V is del Pezzo

surface with canonical singularities.

Suppose that f is a blow up of two F-points O1 and O2 on X. In particular, K3X = 3

and X is a cubic in P3. Let L be a line in P3 passing through points O1 and O2 and HL

be a pencil of hyperplane sections of surface X, which contain line L. Then

dim(Bs(HL)) = 0;

because L 6� X due to Pic(X) = Z. So, equivalence

f�1(HL) � �KW

34

implies that for some integer n � 0 linear system j � nKW j is free and gives birational

map onto del Pezzo surface with canonical singularities.

All other possibilities for morphism f could be handled exactly in the same way as two

cases considered above. �

Note, that surface X may have a lot of non-standart birational transformations to the

del Pezzo surfaces with canonical singularities.

In order to construct elliptic �brations, which are birationally isomorphic to X, I will

introduce one well-known object (see [Do]).

Alphan's Pencil. Pencil P in linear system j�nKX j is called Alphan's pencil, if there

is such commutative diagramW

f . & �j�nKW j

X�P9 9 K P1

;

that K2W = 0 and linear system j � nKW j is free.

De�nition of Alphan's pencil leads to the following agreement.

Agreement. Let P be an Alphan's pencil and commutative diagram

W

f . & g

X�P9 9 K P1

be resolution of indeterminacy of pencil P. I will say that g is an elliptic �bration given

by Alphan's pencil P.

Note, that surface X always has Alphan's pencils.

Example. Every pencil in linear system j � KX j is Alphan's pencil, which gives an

elliptic �bration without multiple �bers.

Before proceeding to the next topic consider the following easy statement.

Statement. In the de�nition of Alphan's pencil curve Z is rational over the �eld F.

Proof. Consider notations of the de�nition of Alphan's pencil. Let n > 1 (otherwise

claim is obvious). Then Riemann-Roch Theorem implies H0(�KW ) = F and unique

e�ective divisor F in linear system j�KW j is a multiple �ber of elliptic �bration �j�nKW j.

35

Thus, action of group Gal(�F=F) on surface W �F leaves divisor F �F invariant. Therefore,

curve Z contains F-point �j�nKW j(F ). �

I will conclude this section with classical descriptions of so-called Bertini and Geizer

involutions of surface X in case K2X 6= 1.

Assumption. K2X = 2.

Suppose that surface X contains such F-point O, that all 56 exceptional curves on the

surface X �F avoid the set O �F and linear system j �KX�Fj does not contain curves,

which are singular in any point of the set O �F .

Let f :W ! X be a blow up of F-point O. Then morphism �j�2KW j is a double cover.

So, let � be an involution interchanging its �bers. Then

��(f�(�KX)) � 3f�(�KX)� 4E and ��(E) � 2f�(�KX)� 3E;

where E = f�1(O).

Bertini Involution of Double Space. Birational involution (O) 2 Bir(X) is called

Bertini involution if it is given by commutative diagram

W�! W

f # # f

X (O)9 9 K X

:

Now consider the case of cubic in P3.

New Assumption. K2X = 3.

Suppose that cubic X contains such irreducible point O, that the following conditions

hold: point O splits into two points over the �eld �F ; linear system j � KX�Fj does not

contains curves passing through both points of set O �F and having singularities in one

of them; cubic X �F does not contain lines passing through any point of the set O �F .

Blow up f :W ! X point O. One can show that morphism �j�2KW j is a double cover.

Let � be an involution, which interchanges �bers of morphism �j�2KW j. Then

��(f�(�KX)) � 5f�(�KX)� 6E and ��(E) � 4f�(�KX)� 5E;

where E = f�1(O).

36

Bertini Involution of Cubic. Involution (O) 2 Bir(X) is called Bertini involution

if it is given by commutative diagram

W�! W

f # # f

X (O)9 9 K X

:

Now I will describe so-called Geizer birational involution of cubic X.

Suppose that cubic surface X has such F-point O, that all lines on cubic surface X �F

avoid point O �F .

Let f :W ! X be a blow up of F-point O. Then �j�KW j is double cover of P2.

Geizer Involution. Involution (O) 2 Bir(X) is called Geizer involution if (O) �ts

commutative diagram

W�! W

f # # f

X (O)9 9 K X

;

where � is an involution interchanging �bers of double cover �j�KW j.

Note, that action on Pic(W ) of involution � in the de�nition of Geizer involution is the

following:

��(f�(�KX)) � 2f�(�KX)� 3E and ��(E) � f�(�KX)� 2E;

where E = f�1(O).

37

x2. Double cover of quadric cone in P3

In this section I will test the technique of Chapter I on the smooth double cover of

two-dimensionl quadric cone.

Object. Let X be a smooth del Pezzo surface over �eld F with Pic(X) = Z and K2X = 1.

In the following points and F-points will mean zero-dimensional schemes and geometri-

cally irreducible points respectively.

Remark. Surface X is biregular to smooth double cover of quadric cone in P3.

Main result of this section is to prove the following theorem.

Double Cone Theorem. There are no birational transformations of surface X into

del Pezzo surfaces with canonical singularities except automorphisms and all birational

transformations of surface X into elliptic �brations are given by Alphan's pencils.

I will reduce Double Cone Theorem to the following statement.

Central Theorem. Singularities of movable log pair (X;MX) with KX +MX �Q 0

are canonical. Moreover, if singularities of log pair (X;MX) are not terminal, then

MX = ��1P (M1P)

for some Alphan's pencil P.

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem imply that surface X is birationally superrigid.

Lemma. Central Theorem implies that X is not birationally isomorphic to any other

del Pezzo surface with canonical singularities.

Proof. Suppose that � : X 9 9 KY is non-trivial birational map of surface X into del

Pezzo surface Y with canonical singularities. Consider log pair

(X;MX) = (X;1

n��1(j � nKY j)) for n� 0:

Then M. N�other & G. Fano & V.A. Iskovskikh's Theorem implies that log pair (X;MX)

is not terminal and Central Theorem conclude the proof. �

Suppose that X is birational to elliptic �bration � : Y ! Z via map �.

Consider log pair

(X;MX) = (X; ��1(j��(D)j)) for deg(D)� 0:

38

Then �(X;MX) = 1. So, M. N�other & G. Fano & V.A. Iskovskikh's Theorem implies that

singularities of log pair (X;MX) are not terminal. Thus, from Central Theorem follows

that linear system ��1(j��(D)j) is composed from some Alphan's pencil.

Observation. Double Cone Theorem is implied by Central Theorem.

In order to prove Central Theorem I will prove �rst one result of [Ma1-2].

Statement on Canonicity of Singularities. In Central Theorem singularities of

movable log pair (X;MX) are canonical.

Proof. Suppose that (X;MX) is not canonical in point O. Then multO(MX) > 1 and

K2X =M2

X � multO(MX)2 > 1:

�

Now I will prove Central Theorem.

Proof of Central Theorem. Consider movable log pair

(X;MX) = (X;nXi=1

biMi)

with KX + MX �Q 0. I may assume that singularities of (X;MX) are canonical and

non-terminal.

Let f :W ! X be a terminal modi�cation of log pair (X;MX). Then log pair

(W;MW ) = (W; f�1(MX))

has terminal singularities and KW +MW �Q 0. In particular, divisor �KW is nef and

equality K2W = 0 holds. Thus,

0 =M2W =

nXi=1;j=1

bibjf�1(Mi) � f

�1(Mj):

The latter implies the claim. �

39

x3. Double cover of P2 ramified in quartic.

This section continues two-dimensional application of Chapter I, which was started in

previous section. cone.

Object. Let X be a smooth del Pezzo surface over �eld F with Pic(X) = Z and K2X = 2.

Let me show one example of such surface X over the �eld Q .

Example. Surface V over Q given by equation

2x40 + 3x41 + 5x42 = x23

of degree 4 in P(1; 1; 1; 2) is smooth del Pezzo surface with K2V = 2 and Pic(V ) = Z.

Proof. Surface V �Q contains exactly 56 exceptional curves and every equation

21

4x0+31

4x1+51

4x2 = 0; 21

4x0+(�3)1

4x1 = 0; 21

4x0+(�5)1

4x2 = 0 and 31

4x1+(�5)1

4x2 = 0;

gives couple of exceptional curves on surface V . On the other hand, group Gal( �Q =Q) acts

on exceptional curves of surface V �Q in such a way, that every orbits of this action contain

at least 3 mutually intersecting curves. Last implies Pic(V ) = Z. �

As before points and F-points will mean zero-dimensional schemes and geometrically

irreducible points respectively.

Remark. X is biregular to double cover of P2 rami�ed in smooth quartic.

Main result of this section is to prove the following theorem.

Double Plane Theorem. Up to the action of Bir(X) all birational transformations of

surface X into del Pezzo surfaces with canonical singularities are standart and all birational

transformations of surface X into elliptic �brations are given by Alphan's pencils.

As in case of Double Cone Theorem (see previous section) Double Plane Theorem is

implied by the following statement.

Central Theorem. Singularities of any minimal (see section 2 of Chapter II) movable

log pair with Q -linearly trivial log canonical divisor are canonical. Moreover, if singularities

of log pair (X;MX) are not terminal, then either

MX = ��1P (M1P)

40

for some Alphan's pencil P, or there is a standart birational transformation � of X into

del Pezzo surface V , such that singularities of log pair (V; �(MX)) are semiterminal.

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem imply that surface X is birationally superrigid.

Lemma. Up to action of group Bir(X) all birational transformations of X into del

Pezzo surface with canonical singularities are standart.

Proof. Suppose that � birationally maps surface X onto del Pezzo surface Y with

canonical singularities.

Consider log pair

(X;MX) = (X;1

n��1(j � nKY j)) for n� 0

with �(X;MX) = 0. We can assume that log pair (X;MX) is maximal.

M. N�other & G. Fano & V.A. Iskovskikh's Theorem implies that singularities of movable

log pair (X;MX) are not terminal. Thus, Central Theorem implies existence of such

birational map � : X 9 9 KV , that log pair

(V;MV ) = (V;1

n� Æ ��1(j � nKY j))

has semiterminal singularities, surface V is del Pezzo surface with canonical singularities

and map � is standart.

Consider � 2 Q>1 , such that log pair (V; �MV ) is canonical. Then movable log pair

(Y;�

n(j � nKY j)

is canonical as well and Uniqueness Theorem implies that birational map � Æ ��1 is an

isomorphism. �

Note, that the rest of Double Plane Theorem can be deduced from Central Theorem

exactly in the same manner as Double Cone Theorem was deduced from the analogeous

statement in previous section.

Observation. Double Plane Theorem is implied by Central Theorem.

Proof of Central Theorem needs the following result of Yu.I.Manin (see [Ma1-2]).

41

Statement on Canonicity of Singularities. In Central Theorem singularities of

minimal movable log pair (X;MX) are canonical.

Proof. Assume, that singularities of minimal movable log pair (X;MX) are worse than

canonical in some point O. Then multO(MX) > 1.

Let

O �F = f �O1; : : : ; �Okg:

Then

K2X =M2

X � multO(MX)2k:

Last inequality implies that k = 1. Therefore, point O is an irreducible F-point.

Consider log pair (X �F ;MX�F) induced by log pair (X;MX) and choose curve C on

surface X �F containing point �O1. Then

�C �KX�F = C �MX�F � multO(MX)mult �O1(C) > mult �O1

(C):

Last implies existence of Bertini involution (O).

Easy calculations show that divisor

MX � (O)(MX)

is ample, which contradicts to maximality of log pair (X;MX). �

Now I will prove Central Theorem.

Proof of Central Theorem. Consider minimal movable log pair

(X;MX) = (X;nXi=1

biMi)

with KX +MX �Q 0. I may assume that log pair (X;MX) is non-terminal canonical.

Consider terminal modi�cation f :W ! X of log pair (X;MX). Log pair

(W;MW ) = (W; f�1(MX))

has terminal singularities and

KW +MW �Q f�(KX +MX) �Q 0:

42

In particular �KW is nef.

Suppose that K2W > 0. Then for big enough integer number n birational map

� = �j�nKW j Æ f�1

is a standart birational transformation of X into del Pezzo surface with canonical singu-

larities V , where V = �j�nKW j(W ). By construction movable log pair (V; �(MX)) has

semiterminal singularities.

Suppose, that K2X = 0. Then

0 =M2W =

nXi=1;j=1

bibjf�1(Mi) � f

�1(Mj)

implies the claim. �

43

x4. Cubic in P3.

In this section I will apply results of Chapter I to smooth cubic surface.

Object. Let X be a smooth del Pezzo surface over �eld F with Pic(X) = Z and K2X = 3.

Let me show one example of such surface X over the �eld Q .

Example. Consider surface V over Q , which is given by equation

3x30 + 5x31 + 7x32 + 11x33 = 0

of degree 3 in P3. Then V is a smooth del Pezzo surface with K2V = 3 and Pic(V ) = Z.

Proof. I will omit the proof, because it is similar to the proof of an analogeous example

in the previous section. �

As before points and F-points stands for zero-dimensional schemes and geometrically

irreducible points respectively.

Remark. X is biregular to smooth cubic surface in P3.

Main result of this section is to formulate the following theorem.

Cubic Theorem. Up to the action of Bir(X) all birational transformations of cubic

surface X into del Pezzo surfaces with canonical singularities are standart and all birational

transformations of surface X into elliptic �brations are given by Alphan's pencils.

Similar to Double Cone Theorem and Double Plane Theorem (see two previous sections)

Cubic Theorem is implied by the following statement.

Central Theorem. Singularities of any minimal (see section 2 of Chapter II) movable

log pair with Q -linearly trivial log canonical divisor are canonical. Moreover, if singularities

of log pair (X;MX) are worse than terminal, then eitherMX = ��1P (M1P) for some Alphan's

pencil P, or there is a standart birational transformation � of surface X into del Pezzo

surface V , such that singularities of log pair (V; �(MX)) are semiterminal.

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem imply that cubic surface X is birationally superrigid.

Proof of Central Theorem. Consider minimal movable log pair

(X;MX) = (X;nXi=1

biMi)

44

with KX +MX �Q 0. Suppose that log pair (X;MX) is not canonical in point O. Then

inequality multO(MX) > 1 holds.

Let O �F = f �O1; : : : ; �Okg. Then

3 = K2X =M2

X � multO(MX)2k:

Thus, k < 2.

Consider movable log pair (X �F ;MX�F) induced by log pair (X;MX) and choose an

arbitrary curve C on surface X �F passing through point �Oi (i 2 [1; k]). Then

�C �KX�F = C �MX�F � multO(MX)mult �Oi(C) > mult �Oi(C):

Last implies existence of either Bertini or Geizer involution (O), such that divisor

MX � (O)(MX)

is ample, which contradicts to maximality of log pair (X;MX).

Therefore, singularities of log pair (X;MX) are canonical and the rest of Central

Theorem can be proved exactly as Central Theorem of previous section. �

45

BIRATIONALLY RIGID FANO 3-FOLDS

x1. Smooth Fano 3-folds.

In this section I will describe well known facts about smooth Fano 3-folds of small degree.

Main Object. Let X be a smooth Fano 3-fold with

Pic(X) = Z and �K3X � 4:

It is not very hard to show that the cube of anticanonical divisor of a smooth Fano

3-fold is an even number. Thus, �K3X is either 2 or 4.

Remark. One can prove (see [Is2]) that

X �=

8>><>>:double cover of P3 rami�ed in sextic in case �K3

X = 2;

quartic 3-fold in P4 in case �K3X = 4;

double cover of smooth quadric in P4 rami�ed in octic in case �K3X = 4:

Birational geometry of X was studied by V.A.Iskovskikh and Yu.I.Manin. They proved

the following result.

Fano 3-fold Theorem. X is birationally rigid and even birationally superrigid if X is

not a double cover of smooth quadric in P4.

In particular, from Fano 3-fold Theorem follows that in case �K3X = 2

Bir(X) = Aut(X):

Note, that Fano 3-fold Theorem implied the following result.

Corollary of Fano 3-fold Theorem. X is not birationally isomorphic to a conic

bundle, �bration on surfaces with Kodaira dimension �1 and any Fano 3-fold with

terminal Q -factorial singularities and Picard group Z, which is not biregular to 3-fold X.

How can 3-fold X be birationally transformed into elliptic �brations or �brations on

surfaces with Kodaira dimension zero? I already described such birational transfomrations

46

for smooth quartic 3-fold (see section 2 of Introduction). However, they could not be

directly generalized for all types of X except for construction of birational transformations

into �brations on K3 surfaces.

Construction of K3 Fibration. Take a pencil P in j�KX j and consider the resolution

of indeterminacy of map �PW

f . & g

X�P9 9 K P1

:

Then g is a �bration on K3 surfaces.

Construction of birational transfomrations of smooth quartic 3-fold into elliptic �bra-

tions can be easily repeated for the double cover of smooth quadric. Nevertheless, in the

next section I will prove the following result.

Non-existence of Elliptic Fibration. X is not birationally isomorphic to an elliptic

�bration in case �K3X = 2.

What can be said about birational transformations of X into arbitrary Fano 3-folds with

canonical singularities? Answer is simple for double cover of P3 and quartic 3-fold.

Uniqueness of Fano 3-fold. Smooth double cover of P3 rami�ed in sextic and smooth

quartic 3-fold could not be non-biregularly birationally transformed into Fano 3-fold with

canonical singularities.

I will prove the last statement in the next two sections.

Surprisingly, if X is a double cover of quadric, then X can be birationally transformed

into Fano 3-fold with canonical singularities, which is not biregular to 3-fold X. I will

conclude this section with the explicit construction of such transformation.

Assumption. I will assume that in the rest of the section �K3X = 4 and X is a double

cover of a smooth quadric Q rami�ed in smooth octic S � Q.

Note, that anticanonical linear system j �KX j is free and morphism �j�KX j is double

cover � : X ! Q.

Lines on X. An irreducible and reduced curve C � X is a line on X if �KX � C = 1.

One can show that X has one-dimensional family of lines, which images on quadric Q

are regular lines.

47

Fix a line C on X.

First Step. Blow up f :W ! X line C and put E = f�1(C).

Denote C1 unique base curve of linear system j �KW j. Then in case �(C) 6� S

�(C) = � Æ f(C1);

otherwise C1 is an exceptional curve of ruled surface E �= F5 (see [Is3]).

Second Step. Blow up g : V !W curve C1 and put G = g�1(C1).

Linear system j �KV j is free and morphism �j�KV j is an elliptic �bration. Moreover,

surface G is a section of �j�KV j. Last implies G �= F1 .

Let C2 be an exceptional section of G and

NC2=V�= OC2

(m)�OC2(n) for m � n:

Then

m+ n = deg(NC2=V ) = c1(V ) � C2 � c1(C2) = �2

and exact sequence

0! NC2=G ! NC2=V ! NC2=V jC2! 0

imlies n � deg(NG=C2) = �1. Therefore,

NC2=V�= OC2

(�1)�OC2(�1):

Third Step. Blow up r : Y ! V curve C2 and put R = r�1(C2).

Note, that R �= P1 � P1.

Forth Step. Contract r : Y ! V surface to the curve C2 � V in such a way, that

birational map r Æ r�1 is not regular.

One can check that r Æ r�1 is a op in curve C2 and r contracts exceptional section of

ruled surface r�1(G) �= F1 . So, G = r Æ r�1(G) is isomorphic to P2 and

NG=V�= OG(�2):

Fifth Step. Contract g : V ! W surface G to point O.

48

Note, that singularities of 3-fold W consist of exactly one terminal point, which is cyclic

quotient singularity of type 12 (1; 1; 1).

I constructed the following diagram

Y

r . & r

V V

g # # g

W W

:

Observation. Map � = g Æ r Æ r�1 Æ g�1 is an anti ip in curve C1.

Note, that map � is a ip for log terminal log pair

(W; (1 + �)j �KW j) for 1� � > 0:

Last implies projectivity and Q -factoriality of 3-fold W .

Base Locus and Degree. �K3W

= 12 and Bs(j �KW j) = O.

Proof. Linear system j �KV j is free and morphism �j�KVj is an elliptic �bration. So,

from construction of map � follows

Bs(j �KW j) = O

and

0 = �K3V= (g�(�KW )�

1

2G)3 = �K3

W�

1

8G3 = �K3

W�

1

2:

�

In particular, divisor �KW is nef and big.

Sixth Step. For big enough n 2 N consider birational morphism �j�nKWj : W ! XC .

Note, that XC is Fano 3-fold with canonical singularities and �K3XC

= 12 .

Corollary. For every line C on 3-fold X there is a birational map

C : X 9 9 KXC ;

such that XC is a Fano 3-fold with canonical singularities and �K3XC

= 12 .

49

x2. Double cover of P3 ramified in sextic.

In this section I will consider smooth Fano 3-fold of the smallest degree. Namely, to the

double cover of P3 rami�ed in smooth sextic.

Object. Let X be a smooth Fano 3-fold with �K3X = 2.

Note, that linear system j �KX j is free and gives double cover � : X ! P3 rami�ed in

smooth sextic S � Q. In particular,

Pic(X) = ZKX and �KX � ��(OP3(1)):

In this section I will prove the following theorem.

Double Space Theorem. 3-fold X is not birationally equivalent to elliptic �brations,

�brations on surfaces with Kodaira dimension zero di�erent from �bration on K3 surfaces

contructed in the previous section and Fano 3-folds with canonical singularities, which are

not biregular to X.

Consider movable log pair (X;MX) with KX +MX �Q 0.

Central Theorem. (X;MX) is canonical, �(X;MX) = 0 and

CS(X;MX) =

(;;

fH1 \H2g for two di�erent surfaces H1 and H2 in j �KX j:

Moreover, if CS(X;MX) 6= ;, then

MX = �1(MP1);

where is a composition of � and projection from the line �(CS(X;MX)).

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem imply that 3-fold X is birationally superrigid. Moreover, just as in the case of

del Pezzo surfaces (see Chapter III) the following can be easily proved.

Important Observation. Central Theorem implies Double Space Theorem.

I will prove Central Theorem in several steps.

First Step. CS(X;MX) does not contain points.

Proof. Suppose CS(X;MX) contains point O 2 X. Take some general enough divisor

HO in linear system j �KX j passing through point O. Then

2 = HO �M2X � multO(M

2X);

50

which contradicts V.A.Iskovskikh & A.V.Pukhlikov's Theorem. �

So, I can assume that set CS(X;MX) contains some irreducible and reduced curve C.

Second Step. �(C) is a line.

Proof. Suppose �(C) is not line. For a general enough divisor H in j �KX j

2 = H �M2X � multC(M

2X)H � C � H � C = �KX �C;

because

multC(MX) � 1:

Thus, �(C) is a conic and �jC is an isomorphism.

Consider blow up f : W ! X of curve C and put E = f�1(C). I want to show that

divisor f�(�3KX)� E is nef.

Suppose �(C) 6� S. Let ~C be such curve on X, that

��1(�(C)) = C [ ~C:

Then

(f�(�2KX)�E) � f�1( ~C) = �2

and

Bs(jf�(�2KX)� Ej) = f�1( ~C):

Thus, f�(�3KX)�E is nef.

Therefore, I may assume that conic �(C) � S. Then

Bs(jf�(�2KX)� Ej) � E:

Let s1 be an exceptional section of ruled surface f jE : E ! C. Then to prove nefness of

divisor f�(�3KX)� E I should prove inequality

(f�(�3KX)�E)jE � s1 � 0:

Elementary properties of blow up implies E3 = 0 and

(f�(�3KX)� E)jE � s1 = 6 +s212:

51

Thus, I have to show that s21 � �12. Let ~S = ��1(S) and

NC=X �= OC(m)�OC(n) for m � n:

Then

m+ n = deg(NC=X) = c1(X) � C � c1(C) = 0

and exact sequence

0! NC= ~S ! NC=X ! N ~S=X jC ! 0

implies n � deg(NC=~S) = �6. So,

s21 = n�m = 2n � �12:

Therefore, divisor f�(�3KX)� E is nef. Consider log pair

(W;MW ) = (W; f�1(MX)):

Then

(f�(3H)� E) �M2W � 0:

On the other hand

(f�(�3KX)�E) �M2W=(f�(�3KX)� E) � (f�(�KX)�multC(MX)E)

2

k

6�multC(MX)(6multC(MX) + 4)

;

which contradicts inequality

6�multC(MX)(6multC(MX) + 4) < 0:

�

Consider pencil HC consisting of surfaces in j�KX j, which contain curve C. Note, that

pencil HC is an inverse image by means of morphism � of pencil of planes in P3 containing

line �(C).

52

Third Step. Equality �KX � C = 2 implies that (X;MX) is canonical in the general

point of curve C and

MX = �1(MP1);

where rational map is a composition of � and projection from the line �(C).

Proof. Resolve indeterminacy of rational map �HC via commutative diagram

W

f . & g

X�HC9 9 K P1

;

such that variety W is smooth, exactly one f -exceptional divisor E lie over general point

of curve C and f is an isomorphism outside of curve C.

Take general �ber D of morphism g. Then D is smooth K3 surface,

D � f�(�KX)� E �kXi=1

aiFi

and for every divisor Fi f(Fi) is a point on curve C.

Consider log pair

(D;MD) = (D; f�1(MX)jD):

Then

MD �Q ((1�multC(MX))E +kXi=1

ciFi)jD

for some rational numbers ci. Thus, multC(MX) = 1 and MD = ;. Last implies the

claim. �

Therefore, I may assume �KX � C = 1.

Forth Step. �(C) 6� S implies (X;MX) is canonical in the general point of C, second

component of ��1(�(C)) belongs to CS(X;MX) and MX = �1(MP1), where rational map

is a composition of � and projection from the line �(CS(X;MX)).

Proof. Let ��1(�(C)) = C [ ~C. Take general enough divisor D in linear system HC .

Then surface D is smooth K3 surface containing curve ~C and

MX jD = multC(MX)C +mult ~C(MX) ~C + RD;

53

where RD is a movable boundary on D.

On D

C2 = ~C2 = �2 and C � ~C = 3:

Hence,

1 =MX jD � ~C = 3multC(MX)� 2mult ~C(MX) + RD � ~C:

So, mult ~C(MX) � 1 and CS(X;MX) contains curve ~C.

Now I can apply arguments of Third Step to the union C [ ~C. �

Fifth Step. �(C) � S implies that log pair (X;MX) is canonical in the general point

of curve C and

MX = �1(MP1);

where is a composition of � and projection from the line �(CS(X;MX)).

Proof. Let f : W ! X be a blow up of curve C and E = f�1(C). Then base locus

of linear system f�1(HC) consists of smooth rational curve ~C, which is a section of ruled

surface f jE : E ! C.

Consider log pair

(W;MW ) = (W; f�1(MX))

and general enough divisor D in linear system f�1(HC). Then D is smooth K3 surface

and

MW jD = mult ~C(MW ) ~C +RD;

where RD { movable boundary.

On the other hand, on surface D ~C2 = �2 and

MW jD �Q ~C + (1�multC(MX))EjD:

Last implies

mult ~C(MW ) = multC(MX) = 1:

Now I can repeat all arguments I used during Third Step to complete the proof. �

Remark. I proved Central Theorem.

54

x3. Quartic 3-fold.

In this section I will apply technique of Chapter I to quartic 3-fold.

Object. Let X be a smooth quartic 3-fold in P4.

Note, that

Pic(X) = Z and�KX � OP4(1)jX :

Main result of this section is the following theorem.

Quartic 3-fold Theorem. Quartic 3-fold X is not birationally isomorphic to the

following varieties: Fano 3-folds with canonical singularities, which are not biregular to X;

elliptic �brations and �brations on surfaces with Kodaira dimension zero except �brations

described in section 2 of Introduction.

Consider movable log pair (X;MX), such that KX +MX �Q 0.

Central Theorem. (X;MX) is canonical, �(X;MX) = 0 and

CS(X;MX) =

8>><>>:;;

fLg; for line L � P4;

fX \Hg; for linear subspace H � P4of dimension 2:

Moreover, if CS(X;MX) 6= ;, then

MX = �1(MY );

where rational map : X 9 9 KY is a projection from CS(X;MX).

Criterion on Superrigidity (see section 2 of Chapter II) and Central Theorem imply

that quartic X is birationally superrigid. Moreover, section 2 of Chapter III implies the

following observation.

Important Observation. Central Theorem implies Quartic 3-fold Theorem.

In the rest of this section I will prove Central Theorem in the similar way as I proved

Central Theorem of the previous section.

First Step. CS(X;MX) does not contain points.

Proof. Suppose CS(X;MX) contains point O. Let HO be general enough hyperplane

section of X passing through point O. Then

4 = HO �M2X � multO(M

2X):

55

Thus, V.A.Iskovskikh & A.V.Pukhlikov's Theorem implies

multO(MX) = 2:

Blow up point O { f :W ! X and put E = f�1(O). Then

a(X;MX ; E) = 0

and

f�1(MX) �Q f�(�KX)� 2E � �KW :

In particular linear system j � nKW j has no �xed components for n� 0.

Let S be inverse image of surface HO on 3-fold W . Then linear system j � KW jSj

contains exactly one e�ective divisor D.

Note, that linear system jnDj is free for n� 0.

On the other hand, D2 = 0 implies

�jnDj(S) = P1:

Therefore, for some some k 2 (1; n] kD is a multiple �ber of �j�nDj. Thus, �j�nDj should

be an elliptic �bration, but arithmetical genus of curve D is 2. �

Thus, I can assume that CS(X;MX) contains irreducible and reduced curve C.

Second Step. Inequality deg(C) � 4 holds.

Proof. C 2 CS(X;MX) implies

multC(MX) � 1:

So, for general hyperplane section H of X

4 = H �M2X � multC(M

2X)H � C � deg(C):

�

Third Step. Curve C is contained in two-dimensional linear subspace in P4.

Proof. Suppose the opposite. Then either C is a smooth curve or C is a rational curve

of degree 4 with one double point P .

56

Suppose C is smooth. Let f :W ! X be a blow up of C, E = f�1(C) and

(W;MW ) = (W; f�1(MX)):

Put

A = (f�(deg(C)H)� E) �M2W :

Then A � 0, because jf�(deg(C)H)�Ej does not have base curves.

On the other hand

A = (deg(C)� deg2(C) + 2g(C)� 2)mult2C(MX)� 2deg(C)multC(MX) + 4deg(C):

Last implies

A � 3deg(C)� deg2(C) + 2g(C)� 2 < 0:

Therefore, C is a rational curve of degree 4 with one double point P . Let f be a

composition of blow up of point P and blow up of inverse image of curve C. Denote G and

E two exceptional divisors of f , where E is an exceptional divisor of blow up of inverse

image of curve C.

Note, P 62 CS(X;MX) and, therefore,

2 > multP (MX) � multC(MX) � 1:

Consider log pair

(W;MW ) = (W; f�1(MX))

and put

A = (f�(4H)�E � 2G) �M2W :

Then A � 0, because linear system jf�(4H)�E � 2Gj does not have basic curves.

On the other hand,

A = �14mult2C(MX) + (4multP (MX)� 8)multC(MX) + 16� 2mutl2P (MX) < 0:

�

57

Forth Step. Suppose C is a line. Then log pair (X;MX) is canonical in general point

of curve C and

MX = �1(MP2);

where rational map is a projection from C.

Proof. Consider linear system HC of hyperplane sections of X passing through line C.

Let f :W ! X be a blow up of curve C, E = f�1(C) and

(W;MW ) = (W; f�1(MX)):

Then

MW + (multC(MX)� 1)E �Q f�1(HC) � �KW ;

linear system j �KW j is free and �j�KW j is an elliptic �bration.

Equality

MW � (f�1(HC))

2 = (1�multC(MX))

implies the claim. �

Thus, I may assume that curve C is not a line and C is contained in unique two-

dimensional linear subspace T � P4. Let HT be a linear system of hyperplane sections

containing plane T .

Fifth Step. deg(C) = 4 implies that log pair (X;MX) has canonical singularities in

the general point of curve C and

MX = �1(MP1);

where rational map is a projection from T .

Proof. Resolve indeterminacy of rational map �HT via commutative diagram

W

f . & g

X�HT9 9 K P1

;

such that 3-fold W contains one divisor E, which dominate curve C, and f is an

isomorphism outside of curve C.

58

Let D be a general �ber of morphism g. Then

D � f�(�KX)� E �kXi=1

aiFi

and for every divisor Fi f(Fi) is a point on curve C.

Consider log pair

(D;MD) = (D; f�1(MX)jD):

For MD the following relation holds

MD �Q ((1�multC(MX))E +kXi=1

ciFi)jD;

where all ci 2 Q .

Thus, movability of log pair (D;MD) implies MD = ; and multC(MX) = 1. Last

implies the claim. �

Therefore, I can assume that C is either a conic or plain cubic.

Sixth Step. All components of X \ T belong to CS(X;MX).

Proof. Tak general enough divisor D in HT . Assume for simplicity, that

X \ T = D \ T = C [rXi=1

Ci;

where all Ci are irreducible and reduced curves on D.

To show

Ci 2 CS(X;MX) for i = 1; : : : ; r

I will prove that intersection form of curves Ci on surface D is negative de�ne.

On surface D

(rXi=1

Ci) � Cj = (DjD � C) � Cj = deg(Cj)� C � Cj :

On the other hand on T

deg(Cj)� C � Cj = deg(Cj)� deg(C)deg(Cj) < 0:

59

Curves Cj are di�erent from C and surface D is smooth. Hence

(C � Cj)D = (C � Cj)T

and [Ar] implies that intersection form of curves Ci on surface D is negative de�ne.

Divisor

MX jD �multC(MX)C �rXi=1

multCi(MX)Ci

is nef on surface D and Q -rationally equivalent to divisor

(1�multC(MX))C +rXi=1

(1�multCi(MX))Ci:

Hence, on surface D

rXi=1

(1�multCi(MX))Ci � Cj � 0 for j = 1; : : : ; r:

Therefore, all multCi(MX) � 1.

Case when curve D\T has multiple component can be considered in the similar way. �

Note, that now I can apply all arguments of the Fifth Step to curve

X \ T:

So, singularities of movable log pair (X;MX) are canonical in the general point of each

component of X \ T and

MX = �1(MP1);

where rational map is a projection from T .

Final Remark. All together Steps implies Central Theorem.

60

x4. Double cover of quadric.

In this chapter I will conclude testing technique of Chapter I.

Object. In the following let

� : X ! Q � P4

be a double cover of a smooth 3-fold quadric Q rami�ed in smooth octic S � Q.

Note, that X is a Fano 3-fold, such that

Pic(X) = �ZKX and �KX � ��(OP4(1)jQ):

Agreement. I will call an irreducible reduced curve C on X a line if �KX � C = 1.

One can easily check that all lines on X are smooth rational curves and their images on

quadric Q are lines in P4 in usual sense.

Remark. For a given line C on X V.A.Iskovskikh constructed

�C 2 Bir(X)

and showed that �C 6� Aut(X) in case �(C) 6� S (see [Is3]).

The following result holds (see [Is3]).

Birational Automorphisms. Bir(X) is generated by �C and Aut(X).

As I said in section 1 of this chapter 3-fold X can be birationally transformed in elliptic

�bration in the same way as quartic 3-fold. Now I will describe such transformations in

details.

Construction of Elliptic Fibration. Choose a line L on Q. Let

: X 9 9 KP2

be a composition of double cover � and projection from L. Consider resolution of the

indeterminacy of rational map W

f . & g

X 9 9 K P2

:

Then g is an elliptic �bration.

61

The following result was proved in section 1 of this chapter.

Fano 3-fold Construction. For every line C on X there is a birational map

C : X 9 9 KXC ;

where XC is singular Fano 3-fold with canonical singularities and �K3XC

= 12 .

In this section I will prove the following theorem.

Double Quadric Theorem. 3-fold X is not birationally isomorphic to the following

varieties: elliptic �brations, which are di�erent from �brations constructed above up to the

action of group Bir(X); �brations on surfaces with Kodaira dimension zero, which are

di�erent from �bration on K3 surfaces constructed in section 1 of this chapter up to the

action of group Bir(X); Fano 3-folds with canonical singularities, which are not isomorphic

to 3-fold X or to 3-fold XC .

Let (X;MX) be an arbitrary minimal (see section 2 of Chapter II) movable log pair,

such that

KX +MX �Q 0:

Then the following holds.

Central Theorem. (X;MX) is canonical, �(X;MX) = 0 and

CS(X;MX) =

8>>>>>><>>>>>>:

;;

fCg; for line C on X;

f��1(L)g; for a line L � Q;

fBs(P)g; for a pencil P in j �KX j:

Moreover, if CS(X;MX) 6= ;, then one of the following holds: MX = �1(MY ), where

: X 9 9 KY is a composition of � and projection from �(CS(X;MX)); for some line C on

X singularities of movable log pair

(XC ;MXC ) = (XC ; C(MX))

are semiterminal and CS(X;MX) = fCg.

Note, that Criterion on Rigidity (see section 2 of Chapter II) and Central Theorem

imply that 3-fold X is birationally rigid.

62

Important Observation. Central Theorem implies Double Quadric Theorems.

In the rest of this section I will prove Central Theorem. Note, that proof of Important

Observation can be found in section 2 of Chapter III.

Exchanging words \hyperplane section" by \general element of anticanonical linear

system" in the proof of First and Second Steps of previous section leads to the following

two assertions.

First Step. CS(X;MX) does not contain points.

Second Step. If CS(X;MX) contains a curve C, then �KX � C � 4.

Therefore, I can assume that locus CS(X;MX) contains an irreducible and reduced

curve C, such that �KX � C � 4.

Third Step. Curve �(C) is either a line or a conic in P4.

Proof. Suppose the opposite. Then �C is an isomorphism and �(C) is either a smooth

rational curve or curve of degree 4 and arithmetical genus 1.

Suppose that �(C) is a smooth rational curve. Let f :W ! X be a blow up of curve C

and E = f�1(C). I want to show that divisor f�(�2KX)� E is nef.

If �(C) 6� S, then

��1(�(C)) = C [ ~C:

Easy to see that

Bs(jf�(�2KX)� Ej) = f�1( ~C):

Thus, divisor f�(�2KX) � E has non-negative intersection with all curves on 3-fold W

except may be f�1( ~C), but

(f�(�2KX)� E) � f�1( ~C) = 0:

If �(C) � S, then Bs(jf�(�2KX)�Ej) � E. Let s1 be an exceptional section of ruled

surface f jE : E ! C. Then nefness of divisor f�(�2KX)� E follows from inequality

(f�(�2KX)�E)jE � s1 � 0:

Note, that E3 = 2� deg(�(C)) and

(f�(�2KX)� E)jE � s1 = 2deg(�(C)) +s21 + 2� deg(�(c))

2:

63

Therefore, nefness of f�(�2KX)�E follows from inequality

s21 � �2� 3deg(�(C)):

Let ~S = ��1(S) and

NC=X �= OC(m)�OC(n) for m � n:

Then

m+ n = deg(NC=X) = c1(X) � C � c1(C) = deg(�(C))� 2

and exact sequence

0! NC= ~S ! NC=X ! N ~S=X)jC ! 0

implies n � deg(NC=~S) = �2� deg(�(C)). So,

s21 = n�m = 2n+ 2� deg(�(C)) � �2� 3deg(�(C)):

Therefore, divisor f�(�2KX)� E is nef. Consider log pair

(W;MW ) = (W; f�1(MX)):

Then

A = (f�(�2KX)�E) �M2W � 0:

On the other hand

A = (f�(�2KX)� E) � (f�(�KX)�multC(MX))

2:

Thus,

A = 8�multC(MX)((2 + deg(�(C)))multC(MX) + 2deg(�(C))) < 0;

because multC(MX) � 1.

Thus, I can assume that �(C) is a curve of degree 4 and arithmetical genus 1. Then

curve �(C) is contained in some hyperplane.

Note, that �KX � C = 4 implies equality multC(MX) = 1.

64

Take general enough divisor H in j �KX j. Divisor H intersects curve C by exactly four

di�erent points: x1, x2, x3 and x4. Let g : V ! H be a blow up of these four points on

surface H and put Ei = g�1(xi). Then

(g�1(MX jH))2 �Q g

�(HjH)�4Xi=1

Ei:

Points �(x1), �(x2), �(x3) and �(x4) are contained in one two-dimensional plane, but

their are not contained in one line. Last implies that linear system

jg�(HjH)�4Xi=1

Eij

contains exactly one e�ective divisor D.

On the other hand, jnDj does not have �xed components for n� 0 and D2 = 0. Thus,

for n� 0 linear system jnDj is free and

�jnDj(V ) = P1:

Hence, for k 2 (1; n] �bration �jnDj has multiple �ber kD. Therefore, arithmetical genus

of curve D should be equal to 1, but arithmetical genus of curve D is 4. �

Forth Step. Suppose that �(C) is a conic. Then log pair (X;MX) is canonical in

general point of curve C and

MX = �1(MP1);

where rational map : X 9 9 KP1 is composition of � and projection from �(C).

Proof. Consider pencil HC in linear system j �KX j, which consists of surfaces passing

through curve C.

Suppose that �KX �C = 4 and resolve indeterminacy of rational map �HC via commu-

tative diagramW

f . & g

X�HC9 9 K P1

:

I may assume that 3-fold W is smooth, W contains exactly one f -exceptional divisor E

lying over general point of curve C and f is an isomorphism outside of C.

65

For general �ber D of g

D � f�(�KX)� E �kXi=1

aiFi;

where f(Fi) is a point for every divisor Fi.

Consider log pair

(D;MD) = (D; f�1(MX)jD):

Then

MD �Q ((1�multC(MX))E +kXi=1

ciFi)jD

and all ci 2 Q . Hence, multC(MX) = 1 and MD = ;. Last implies the claim.

So, �KX �C = 2. Suppose �(C) 6� S and put

��1(�(C)) = C [ ~C:

It is enough to show that CS(X;MX) contains curve ~C. Because then I can apply

previous arguments to reducible curve C [ ~C.

Take general enough divisor D in HC . Then ~C � D and

MX jD = multC(MX)C +mult ~C(MX) ~C + RD

for movable boundary RD.

On surface D

C2 = ~C2 = �2 and C � ~C = 4:

Thus,

2 =MX jD � ~C = 4multC(MX)� 2mult ~C(MX) + RD � ~C:

So, mult ~C(MX) � 1 and CS(X;MX) contains curve ~C.

Thus, I have to consider the case �KX � C = 2 and �(C) � S.

Let f : W ! X be a blow up of curve C and f�1(C) = E. Then base locus of linear

system f�1(HC) consists of a section of ruled surface fE : E ! C. I will denote this

section by ~C.

66

Consider log pair

(W;MW ) = (W; f�1(MX)):

Let D be a general enough surface in linear system f�1(HC). Then

MW jD = mult ~C(f�1(MX) ~C +RD;

where RD { movable boundary.

On the other hand on surface D

f�1(MX)jD �Q ~C + (1�multC(MX))EjD

and ~C2 = �2. Therefore,

mult ~C(MW ) = multC(MX) = 1:

Now I can repeat all arguments of the case �KX �C = 4 and conclude the proof. �

Fifth Step. Suppose that �KX � C = 2 and �(C) is a line. Then singularities of

movable log pair (X;MX) are canonical in the general point of curve C and

MX = �1(MP2);

where rational map : X 9 9 KP2 is composition of � and projection from �(C).

Proof. Consider linear system HC of surfaces in j � KX j containing curve C and

resolution of indeterminacy f :W ! X of rational map �HC . Then claim follows from the

equality

f�1(MX) � (f�1(HC))

2 = 1�multC(MX):

�

Sixth Step. Log pair (X;MX) is canonical.

Proof. Suppose the opposite. Previous Steps imply that log pair (X;MX) is not

canonical in general point of a line C on X. In particular,

multC(MX) > 1:

67

Suppose �(C) 6� S. Then [Is3] implies existence of �C 2 Bir(V ), such that

KX +1

9� 8multC(MX)�C(MX) �Q 0:

Last contradicts minimality of log pair (X;MX).

Suppose now that �(C) � S. Blow up f :W ! X curve C and put

(W;MW ) = (W; f�1(MX)):

Then MW has negative intersection with inverse images on W of all lines on Q, which are

tangents to surface S in the points of curve �(C). Last contradicts to movability of log

pair (W;MW ). �

Now I am ready to prove Central Theorem.

Proof of Central Theorem. I may assume that singularities of (X;MX) are canonical

and locus CS(X;MX) contains a line C. Blow up f :W ! X curve C and put E = f�1(C).

Suppose that log pair

(W;MW ) = (W; f�1(MX))

is terminal. I claim that singularities of log pair

(XC ;MXC ) = (XC ; C(MX))

are semiterminal.

Let us brie y remind construction of birational map C (see section 1 of this chapter).

Blow up f : W ! X curve C and denote C1 unique base curve of linear system j �KW j.

Then make anti ip � :W 9 9 KW in curve C1 and put

C = �j�nKWj Æ �:

Take � 2 Q>1 , such that log pair (W; �MW ) is still terminal. Then � is log ip for log

pair (W; �MW ). In particular, log pair

(W ; �MW ) = (W; �� Æ f�1(MX))

68

is terminal. On 3-fold W the following relation holds

KW +MW �Q 0:

Therefore, morphism �j�nKWj is crepant for log pair (W ; �MW ). Last implies canonicity

of log pair (XC ; �MXC ). So, log pair (XC ;MXC ) is semiterminal by de�nition.

Therefore, in order to prove the claim I may assume that log pair (W;MW ) is not

terminal.

Note, that (W;MW ) is a crepant pull back of log pair (X;MX). Thus, I may assume

that set CS(W;MW ) contains a smooth rational irrreducible curve T , such that curve f(T )

is a line on 3-fold X.

There are four cases:

T = C1; � Æ f(T ) \ �(C) = ;; f(T ) = C and � Æ f(T ) \ �(C) 6= ;:

In case T = C1 arguments of the proof of the Fifth Step implies the claim.

Suppose that � Æ f(T ) \ �(C) = ;. Take general surface DW in anticanonical linear

system j �KW j. Then DW is a smooth K3 surface, T \DW 6= ; and

MW jDW �Q C1 + F;

where F is elliptic curve, such that F � C2 = 1.

On the other hand

MW jDW = multC1(MW )C1 + RDW ;

for movable boundary RDW . Moreover, multC1(MW ) > 0, because

MW jDW � C1 = �1:

Taking intersection of MW jDW with a curve in pencil jF j passing through any point of

set T \DW leads to a contradiction.

Suppose that f(T ) = C. Then T � E and I may assume T 6= C1.

In case C1 � E intersecting MW with a �ber of ruled surface E leads to E �MW . The

latter contradicts to movability of log pair (W;MW ).

69

In case C1 6� E one can easily show that

T \DW 6= ;;

where DW is a general enough divisor in linear system j �KW j. Therefore, argumens of

the case � Æ f(T ) \ �(C) = ; leads to contradiction.

So, f(T ) 6= C and to conclude the proof I may assume that � Æ f(T ) \ �(C) 6= ;.

Take a pencil HT � j�KW j consisting of surfaces passing through curve T . Note, that

pencil HT is an inverse image on W of a pencil of hyperplane sections of quadric Q passing

through lines � Æ f(C) and � Æ f(T ).

Let h : U ! W be a blow up of curve T . On 3-fold U base locus of pencil j � KU j

consists of two smooth irreducible curves. By construction one of these curves is h�1(C1),

denote the other one as T1.

Let DU be a general enough surface in j �KU j. Then DU is a smooth K3 surface.

On DU

h�1(MX)jDU �Q h�1(C1) + T1

and

h�1(C1)2 = T 2

1 = (h�1(C1) + T1)2 = �2:

Hence,

multC1(MW ) = multT1(MW ) = 1:

Now argumens of the proof of Forth Step leads to

CS(X;MX) = fBs(f(HT ))g and MX = ��1f(HT )(MP1):

�

70

CONIC BUNDLES

x1. Surface conic bundles.

In this section I will apply results of Chapter I to smooth two-dimensional conic bundles

over an arbitrary �eld F with algebraic closure �F .

Note, that all results of Chapters I and II are valid over �eld F in two-dimensional case.

As in Chapter III points and F-points will mean zero-dimensional schemes and geomet-

rically irreducible points respectively.

Object. Fix smooth two-dimensional conic bundle � : X ! S with

Pic(X=S) = Z and K2X < 0:

Note, that inequality K2X < 0 implies that �eld F is not algebraically closed.

Observation. K2X < 0 () morphism � has more then 8 degenerate �bers.

V.A.Iskovskikh proved in [Is1] that conic bundle

� : X ! S

is birationally superrigid in a weaker assumption that K2X � 0. Purpose of this section is

to prove the following result.

Surface Conic Bundle Theorem. X is not birationally isomorphic to del Pezzo

surfaces with canonical singularities and to elliptic �brations.

In order to prove Surface Conic Bundle Theorem consider movable log pair

(X;MX);

such that KX +MX �Q ��(L), where L is a divisor on curve S.

Observation. deg(L) > 0.

Proof. Let deg(L) � 0. Then 0 > K2X = �MX � ��(L) +M2

X � �MX � ��(L) � 0. �

Moreover, the following result holds.

71

Central Theorem. I(X;MX) = � and �(X;MX) = 1.

Note, that Criterion on Fibration Superrigidity (see section 2 of Chapter II) together

with Central Theorem implies that � : X ! S is birationally superrigid.

Observation. Central Theorem implies Surface Conic Bundle Theorem.

Proof. Suppose that � : X 9 9 KY is non-trivial birational map of surface X into del

Pezzo surface Y with canonical singularities. Consider log pair

(X;MX) = (X;�

n��1(j � nKY j)) for n� 0 and � 2 Q>0 ;

such that KX +MX is Q -rationally �-trivial. Then �(X;MX) = 1 by Central Theorem.

On the other hand, due to construction

�(X;MX) =

8>><>>:�1 for � < 1;

0 for � = 1;

2 for � > 1:

Suppose that surface X is birationally equivalent to elliptic �bration

� : Y ! Z

by means of some birational map �. Consider log pair

(X;MX) = (X; ��1(j��(D)j)) for deg(D)� 0:

Then due to construction �(X;MX) = 1 and I(X;MX) = � Æ �. On the other hand, one

can easily show that Central Theorem implies I(X;MX) = �. �

Consider one class of birational maps.

Good Birational Maps. Let U be a class of all birational maps

: X=S 9 9 KW=S;

such that surface W is smooth, Pic(W=S) = Z and equality K2X = K2

W holds.

Note, that class U is not empty.

Auxiliary Theorem. For some � 2 U singularities of (�(X); �(MX)) are canonical.

72

Proof. Consider the following function

q( ) = #CS( (X); (MX)) for 2 U :

Take birational map � 2 U minimizing q and put

(Y;MY ) = (�(X); �(MX))

I claim that � is desired birational map. Indeed, suppose that (Y;MY ) is not canonical

in some point y 2 Y . I will show that the latter leads to a contradiction.

Let

�Y = Y �F ; �S = S �F ; M�Y =MY �F

and �� : �Y ! �S be a morphism induced by � . Suppose

y �F = f�y1; : : : ; �ykg:

Then multiplicity of boundary M �Y in every point �yi is strictly bigger than 1.

Take �ber �Fi of morphism �� containing point �yi. I want to show that �Fi is irreducible

and does not contain point �yj if j 6= i.

Firstly,

2 = �BY � �Fi =X�yj2 �Fi

( �BY � �Fi)�yj �X�yj2 �Fi

mult�yj ( �MY )mult�yj ( �Fi) > #f�yj 2 �Fig:

Hence, �ber �Fi contains exactly one point among f�y1; : : : ; �ykg, which is a smooth point of

�ber �Fi.

Secondly, Pic(Y=S) = Z implies non-existence of irreducible components of the reducible

�bers of morphism �� , which are invariant via the action of Gal(�F=F). Therefore, if �ber

�Fi is reducible, then �Fi has to contain at least two points among

f�y1; : : : ; �ykg;

but as I already proved it is impossible.

73

Consider �ber F of morphism � containing point y. I can de�ne birational map

� : Y=S 9 9 KV=S

as a composition of blow up point y and blow down the strict transform of �ber F . Then

K2V = K2

X and q(� Æ �) < q(�):

�

Auxiliary Theorem implies Central Theorem. Therefore, Surface Conic Bundle Theorem

is �nally proved.

Example. Let W � P1 � P2 be a surface over Q given by the zeroes of the following

polynomial of bi-degree (5; 2):

t0(4t21� t

20)(t

20� 25t

21)x

20+(4t1� t0)(t

20� 9t

21)(t

20� t

21)x

21� (t0+4t1)(t

20� 36t

21)(t

20� 49t

21)x

22:

Then surface W is smooth, K2W = �7 and projection to P1 gives a conic bundle with

exactly �ve reducible �bers. Contracting exceptional curves leads to smooth conic bundle

� : X ! P1 with Pic(X=P1) = Z and K2X = �2.

74

x2. 3-fold conic bundles.

In this section I will consider three-dimensional birationally rigid conic bundles.

Object. Fix 3-fold Mori �bration � : X ! S with dim(X=S) = 1.

Consider non-movable log pair

(S;1

4DS);

where DS is a degeneration divisor of �.

Main Assumption. Log pair (S; 14DS) has canonical singularities and

�(X;1

4DS) = 2:

Note, that if conic bundle � is standard (see [Sa1-2]), then log pair (S; 14DS) has canonical

singularities.

Example. Let f : V ! P2 be a natural projection of 4-fold

V = Proj(OP2(�5)�OP2(�5)�OP2)

and 3-fold X be a general divisor in linear system jOV=P2(2)+ f�(OP2(11))j. Then induced

morphism f jX : X ! P2 is a conic bundle, 3-fold X is smooth, Pic(X=P2) �= Z,

degeneration divisor DS of f jX has only double points and deg(DS) = 13 (see [Be]).

V.G.Sarkisov proved in [Sa1-2] birational superrigidity of conic bundle

� : X ! S

even in a weaker assumption that �(X; 14DS) � 0. Purpose of this section is to prove the

following result.

3-fold Conic Bundle Theorem. X is not birationally isomorphic to Fano 3-folds

with canonical singularities, elliptic �brations and �brations on surfaces, whose Kodaira

dimension is zero.

Consider log pair (X;MX) such that

KX +MX �Q ��(L);

75

where L is a Q -Cartier divisor on surface S.

From the arguments of the previous section follows that 3-fold Conic Bundle Theorem

is implied by the following result.

Central Theorem. I(X;MX) = � and �(X;MX) = 2.

Note, that Criterion on Fibration Superrigidity (see section 2 of Chapter II) together

with Central Theorem implies birational superrigidity of � : X ! S.

In the rest of this section I will prove the following result, which implies Central Theorem.

Theorem. There is a commutative diagram

X�9 9 K Y

� # # �

S�9 9 K Z

;

such that maps � and � are birational, � is a Mori �bration, log pair

(Y;MY ) = (Y; �(MX))

has canonical singularities and

KY +MY �Q ��(H);

where H is nef and big Q -Cartier divisor on Z.

A.Corti in [Co1] proved the existence of commutative diagram

X 9 9 K X

� # # �

S� S

;

such that map and morphism � are birational, � is Mori �bration and log pair

(X;MX) = (X; (MX))

is canonical.

Observation. Singularities of surface S are log terminal.

76

Let DS be a degeneration divisor of conic bundle �.

Lemma I. �(KS +14DS) = 2.

Proof. Divisor DS � ��1(DS) is e�ective. So, canonicity of (S; 14DS) implies the

e�ectiveness of some multiple of divisor

KS +1

4��1(DS)� �

�(KS +1

4DS):

Therefore, for n� 0

h0(n(KS +1

4DS)) � h

0(n(KS +1

4��1(DS))) � h

0(n(KS +1

4DS)):

�

For some Q -Cartier divisor L on surface S

KX +MX �Q ��(L):

Lemma II. The following relation holds

L �Q KS +1

4DS +

1

4��(M

2X):

Proof. Well-known (see [Sa1-2]) that

���(K2X) �Q KS +

1

4DS :

Claim follows from

��(L)2 � 2KX � ��(L) �Q K

2X:

�

Proof of Theorem. Consider divisor L as as a log canonical divisor of log pair

(S;1

4DS +

1

4��(M

2X)):

Then [KeMaMc] implies existence of such boundary BS on surface S, that

BS �Q1

4DS +

1

4��(M

2X)

77

and singularities of (S; BS) are log terminal. Thus,

KX +MX �Q ��(KS + BS):

If divisor KS + BS is nef, then Log Abundance Theorem (see [KeMaMc]) implies the

claim. Therefore, I can assume that divisor KS +BS is not nef.

Log Minimal Model Program implies the existence of birational morphism

q : S ! ~S

contracting one irreducible reduced curve C on surface S, such that

(KS + BS) � C < 0:

Put B ~S = q(BS). Then log pair ( ~S;B ~S) is log terminal and

KS + BS �Q q�(K ~S + B ~S) + aC

for some a 2 Q>0 .

Application of relative Log Minimal Model Program over surface ~S to (X;MX) gives

commutative diagram

Xp9 9 K ~X

� # # ~�

Sq! ~S

;

such that map p is birational and divisor K ~X +M ~X is ~�-nef, where

( ~X;M ~X) = ( ~X; p Æ (MX)):

Note, then ~X has terminal Q -factorial singularities and

K ~X +M ~X �Q ~��(K ~S + B ~S) + p(aG);

where G = ��1(C).

Note, that ~�-nefness of log canonical divisor K ~X +M ~X implies that G is contracted by

birational map p. So, ~� is a Mori �bration,

K ~X +M ~X �Q ~��(K ~S + B ~S)

and

�(K ~S +M ~S) = �(KS + BS) = 2:

Repeating our construction at most rk(Pic(S)) times we obtain the claim. �

78

FANO HYPERSURFACES

x1. Hypersurfaces of degree N in PN .

In this section I will consider applications of results of Chapter I to general enough

hypersurfaces of degree N in PN in case when N is greater than 4.

Object. Let X be a general enough hypersurface of degree N in PN for N � 5.

Note, that

Pic(X) = �ZKX and�KX � OPN(1)jX :

A.V.Pukhlikov in [Pu5] proved that X is birationally superrigid. In particular, X is not

birationally equivalent to any �brations on varieties of smaller dimension, whose Kodaira

dimension is �1.

In which �brations on varieties with Kodaira dimension zero can be birationally

transformed hypersurface X?

Construction of Calabi-Yau Fibration. Take any pencil P of hyperplane sections

of X and resolve the indeterminacy of rational map �P by means of commutative diagram

W

f . & g

X�P9 9 K P1

:

Then g is a �bration on varieties with Kodaira dimension zero.

Main result of this section is the following theorem.

Hypersurface Theorem. X is not birationally isomorphic to �brations on varieties

with Kodaira dimension zero except �brations described above.

Consider movable log pair (X;MX) with

KX +MX �Q 0:

From the methods of Chapters III and IV follows that Hypersurface Theorem is implied

by the following statement.

79

Central Theorem. Singularieties of (X;MX) are canonical, �(X;MX) = 0,

CS(X;MX) =

(;;

X \H for a linear subspace H � PN of dimension N � 2:

Moreover, if CS(X;MX) 6= ;, then

MX = ��1P (MP1);

where P is a pencil in j �KX j, such that CS(X;MX) = fBs(P)g:

In the rest of the section I will prove Central Theorem.

First Step. CS(X;MX) does not contains points.

Proof. Follows from [Pu5] and V.A.Iskovskikh & A.V.Pukhlikov's Theorem. �

Thus, I can assume that set CS(X;MX) contains some subvariety S � X of non-zero

dimension. Moreover, paper [Pu3] implies that singularities of movable log pair (X;MX)

are canonical. In particular, multS(MX) = 1.

Second Step. dim(S) = N � 3.

Proof. Let f :W ! X be a blow up of a general point5 of S. Then

a(X;MX ; E) = N � 2� dim(S)�multS(MX) = N � 3� dim(S)

and inequality dim(S) < N � 3 implies existence of subvariety T � E, such that induced

morphism f jT : T ! S is a surjection and

T 2 CS(W; f�1(MX)� a(X;MX ; E)E):

In particular, multS(MX) � multT (f�1(MX)) > 1. �

Third Step. deg(S) � N .

Proof. N = (�KX)N�3 �M2

X � multS(M2X)deg(S). �

Note, that deg(S) 6= 1 due to generality of X.

Forth Step. S is contained in linear subspace of PN of dimension N � 2.

5Variety W is quasiprojective.

80

Proof. Intersecting X with general enough hyperplane sections I can assume that X

is a hypersurface of degree N in P4 containing curve S. Using main trick of paper [Pu2] I

will show that curve S is contained in some two-dimensional linear subspace of P4.

Consider general cone RS over curve S. Then

RS \X = S [ ~S

and deg( ~S) = (N � 1)deg(S).

Let

Z = Supp([ni=1Bs(Mi)):

Then S � Z, but generality of RS implies ~S 6� Z.

A.V.Pukhlikov showed in [Pu2], that generality of RS implies curves S and ~S intersect

transversally in (N � 1)deg(S) di�erent points. On the other hand,

deg( ~S) = deg(MX j ~S) � (N � 1)deg(S)multS(MX) = (N � 1)deg(S):

So,

~S \MX = S \ ~S:

Note, that general secant of curve S intersects X exactly in N points, because otherwise

it is contained in X and should be a component of Z.

Consider divisor

D =nXi=1

biMi;

where Mi is a general member of linear systemMi. Then by assumption

multS(D) = 1:

Choose two general enough points PS and PD on curve S and divisor D respectively.

Let L be a line passing through points PS and PD and P be a general enough point on L.

Denote RS;P a cone over curve S with a vertex P and

RS;P \X = S [ ~SP :

81

Note, that I already shown before that divisor D either contains curve ~SP or intersects

it only in points S \ ~SP .

By construction

PD 2 ~SP \D and PD 62 S:

Hence, ~SP � D and, in particular,

L \X � D:

Last condition is closed and I can assume

PD 2 SnPS:

Therefore, due to generality of divisor D general enough secant of curve S should intersect

Z in N di�erent points. On other hand, let A be a set of 3 di�erent collinear points in

general hyperplane section of Z. Then one can show that

A � Z 0;

where Z 0 is a component of Z, such that Z 0 is contained in 2-dimensional linear subspace

of P4. �

Note, that I never used irreducibility of variety S. Therefore, I can assume in the

following that variety S is a union of all elements of CS(X;MX). So, S is contained in

unique linear subspace

T � PN

of dimension N � 2. Denote HT a pencil on hypersurface X cutted by hyperplane sections

of X containing T .

Fifth Step. Equality deg(S) = N implies Central Theorem.

Proof. Consider resolution of the indeterminacy of rational map �HT by means of such

birational morphism f :W ! X, that variety W is smooth. Put

(W;MW ) = (W; f�1(MX)):

Let g = �HT Æ f and D be general enough �ber of g. Then

MW jD �Q

kXi=1

aiFi;

82

where all ai are positive rational numbers and codimension of all subvarieties

f(Fi) � X

are greater than 2.

Movability of MW jD implies that the boundary MW lies in the �bers of g. Therefore,

for some boundary MP1 on P1

MX = ��1P (MP1):

�

Let

X \ T = S [rXi=1

Si;

where Si are irreducible and reduced subvarieties of X of dimension N � 3. Then to prove

Central Theorem I have to show that CS(X;MX) contains all Si.

Sixth Step. CS(X;MX) contains all subvarieties Si � X.

Proof. Intersecting X with general enough hyperplane sections I may assume that X

is a hypersurface of degree N in P4, T is two-dimensional linear subspace in P4, S and all

Si are curves.

Take a surface D of pencil HT , which is smooth in the points of intersection of curve S

with curves Si. On D

(rXi=1

Si) � Sj = (DjD � S) � Sj = deg(Sj)� S � Sj :

On plane T

deg(Sj)� S � Sj = deg(Sj)� deg(S)deg(Sj) < 0:

So,

(S � Sj)D = (S � Sj)T ;

because surface D is smooth in the points of intersection of curve S with curves Si. From

paper [Ar] follows that intersection form of curves Si on surface D is negative de�ned.

Divisor

MX jD � S �rXi=1

multSi(MX)Si

83

is nef on surface D. On other hand, on D

MX jD � S �rXi=1

multSi(MX)Si �Q

rXi=1

(1�multSi(MX))Si

andrXi=1

(1�multSi(MX))Si � Sj � 0 for j = 1; : : : ; r:

Negativity of intersection form of curves Si on surface D implies

multSi(MX) � 1

for all Si. �

84

x2. Quintic 4-fold.

In this section I will apply results of Chapter I to any smooth quintic 4-fold in P5.

Object. Let X be a smooth quintic 4-fold in P5.

Note, that

Pic(X) = �ZKX and�KX � OP5(1)jX :

In 1987 A.V.Pukhlikov strengthening method of V.A.Iskovskikh and Yu.I.Manin proved

in paper [Pu1] that quintic X is birationally superrigid.

In which elliptic �brations and �brations on surfaces and 3-folds with Kodaira dimension

zero can be birationally transformed smooth quintic X? Previous section contains a con-

struction of birational transformation of X into �brations on Calabi-Yau 3-folds.

Construction on K3 Fibration. Suppose that quintic X contains two-dimensional

linear subspace P � P5. Let : X 9 9 KP2 be a projection from P and diagram

W

f . & g

X 9 9 K P2

be a resolution of the indeterminacy of map . Then g { �bration on K3 surfaces.

Note, that general quintic 4-fold in P5 does not contain 2-dimensional planes.

Purpose of this paper is to give an alternative proof of birational superrigidity of X and

prove the following statement.

Quintic Theorem. Quintic X is not birationally isomorphic to �brations on varieties

with Kodaira dimension zero except for �brations on Calabi-Yau 3-folds and K3 surfaces

described above.

Fix movable log pair (X;MX) such that

KX + �MX �Q 0:

Methods of Chapters III and IV implies that to prove Quintic Theorem it is enough to

prove the following statement.

85

Central Theorem. Singularities of (X;MX) is canonical, �(X;MX) = 0 and

CS(X;MX) =

8>><>>:;;

fSg; for linear subspace S � X of dimension 2;

fX \Hg; for linear subspace H � P5 of dimension 3:

Moreover, in case CS(X;MX) 6= ;

MX = �1(MY );

where rational map : X 9 9 KY is a projection from CS(X;MX).

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem implies that quintic X is birationally rigid.

Statement. Central Theorem is implied by the fact that set CS(X;MX) does not

contain points.

Proof. Arguments of the previous chapter work for quintic X except the case

when CS(X;MX) contains point or when CS(X;MX) contains only 2-dimensional linear

subspace T in P5.

Consider two-dimensional linear system HT of hyperplane sections of quintic X passing

through T and blow up f : W ! X plane T . Let g = �HT Æ f and S be general enough

�ber of �bration g. Then

f�1(MX) � S = 0

Thus, f�1(MX) lies in the �bers of g �

Suppose that CS(X;MX) contains point O. In the rest of the section I will deduce a

contradiction with this assumption.

Consider log pair

(X;BX) = (X;HX +MX);

where HX is a general enough hyperplane section of X passing through point O.

Observation. O 2 LCS(X;BX).

Let f :W ! X be a blow up of O and E = f�1(O). Then

a(X;BX ; E) = a(X;MX ; E)� 1:

86

Lemma I. The following inequalities hold

a(X;BX ; E) > �1 and a(X;MX ; E) > 0:

Proof. Suppose the claim is not true. Then multO(MX) � 3 and

5 = (�KX)2 �M2

X � multO(M2X) � mult

2O(MX) � 9:

�

Consider log pair

(W;BW ) = (W; (multO(MX)� 2)E + BW ) = (W; (multO(MX)� 2)E +HW +MW );

where HW = f�1(HX) and MW = f�1(MX).

Observation. By construction

KW + BW �Q f�(KX +BX):

The following lemma is a corollary of Lemma I.

Lemma II. LCS(W;BW ) contains proper irreducible subvariety of E, which is not

contained in HW .

Proof. Equivalence

(multO(MX)� 3)E +MW �Q f�(KX +MX)

and Lemma I imply

S 2 CS(W; (multO(MX)� 3)E +MW );

where S is a proper irreducible subvariety of E.

Therefore,

S 2 LCS(W;multO(MX)� 2)E +MW )

and generality of HX completes the proof. �

87

Now I will consider a subvariety of W , that will play rather important role later.

New Object. Let S be an element of maximal dimension of LCS(W;BW ), such that

variety S is a proper subvariety of E and S 6� HW .

Note, S can be a point, curve or surface.

Lemma II. S is not a surface.

Proof. Suppose the opposite. The fact, that

S 2 LCS(W; (multO(MX)� 2)E +MW )

allows me to apply A.Corti's Lemma (see section 3 of Chapter I). Last gives

multS(M2W ) � 4(3�multO(MX)):

Thus,

multO(M2X) � mult

2O(MX) +multS(M

2W ) � mult2O(MX) + 4(3�multO(MX)):

Hence,

5 = (�KX)2 �M2

X � multO(M2X) � (multO(MX)� 2)2 + 8:

�

Lemma IV. S is not a point.

Proof. V.V.Shokurov's Relative Connectedness Theorem (see section 3 of Chapter I)

implies connectedness of locus LCS(W;BW ) in the neighbourhood of divisor E. The latter

implies the claim. �

The following lemma is a Corollary 3.6 of [Co2].

Lemma V. S is a \line" in E �= P3.

Proof. Connectedness of

LCS(W; (multO(MX)� 2)E +HW +MW )

in the neighbourhood of E and generality in the choice of HX together with adjunction

formula imply that

LCS(HW ; (multO(MX)� 2)EjHW +MW jHW )

88

contains only points and

fS \HWg � LCS(HW ; (multO(MX)� 2)EjHW +MW jHW ):

Applying V.V.Shokurov's Relative Connectedness Theorem to locus

LCS(HW ; (multO(MX)� 2)EjHW +MW jHW )

and morphism f jHW leads to connectedness of the losus

LCS(HW ; (multO(MX)� 2)EjHW +MW jHW )

in the neighbourhood of EjHW . So, S \HW consists of one point. �

Note, that Inverse of Adjunction (see section 3 of Chapter I) implies

O 2 LCS(HX ;MX jHX ):

Unfortunately, this is not enough to get a contradiction.

Let Y be a general enough hyperplane section of X passing through O, such that

S � f�1(Y ):

Put

(Y;MY ) = (Y;MX jY ):

Warning. Y may be singular and (Y;MY ) may no longer be movable!

Nevertheless, I can handle log pair (Y;MY ) in the neighbourhood of point O.

Remark. Point O is smooth on Y and

O 2 LCS(Y;MY ):

Let g : V ! Y be a blow up of O and F = g�1(O). Identify V with a subvariety of W ,

by construction

S � F; EjV = F; multO(MY ) = multO(MX) and g�1(MY ) =MW jV :

89

Consider log pair

(V;MV ) = (V; (multO(MY )� 2)F +MV );

where MV = g�1(MY ). Then

KV +MV �Q f�(KY +MY ):

Most Important Property of Y . Curve S belongs to LCS(V;MV ).

Proof. By construction S � V and

S 2 LCS(W;BW ):

So, log pair (V;MV ) is not log terminal in the neighbourhood of curve S due to adjunction

formula and V.V.Shokurov's Relative Connectedness Theorem.

Suppose that LCS(V;MV ) does not contain S. Then LCS(V;MV ) contains some

point on curve S. Moreover, from application of V.V.Shokurov's Relative Connectedness

Theorem to (V;MV ) follows that LCS(V;MV ) contains exactly one point on S.

Let h : U !W be a blow up of curve S and G = h�1(S). Consider log pair

(U;BU) = (U; h�1(BW ) + (multS(BW )� 2)G):

Then

KU +BU �Q h�(KW + BW ):

Therefore, adjunction formula and V.V.Shokurov's Relative Connectedness Theorem

imply (see proof of Lemma I) existence of curve

~S � G

in LCS(U;BU), such that ~S is a section of hjG and either curve ~S is contained in h�1(V )

or intersection of ~S and h�1(V ) consists of one point.

Note, that everything is local with respect toX. So, application of Kawamata-Viehweg's

Vanishing Theorem (see [KaMaMa]) to divisor h�1(V )�G leads to surjectivity

H0(h�1(V ))! H0(h�1(V )jG)! 0:

90

On the other hand, linear system jh�1(V )jGj is free and, therefore, due to generality in

the choice of Y curve ~S is not contained in h�1(V ).

Direct calculations show that divisor h�1(V )jG is nef and big on G. Moreover, its

intersection with any section of morphism hjG is either trivial or consists of more than

one point, but I already proved that inersection of ~S and h�1(V ) consists exactly of one

point. �

Lemma VI. Log pair (Y;MY ) is non-movable.

Proof. If (Y;MY ) is movable, then I can literally repeat the proof of Lemma III and

obtain a contradiction. �

Quintic X contains 2-dimensional plane P , such that S � f�1(P ), because otherwise

log pair (Y;MY ) is movable.

Note, that multipicity of P in MY is multP (MX).

Splitting. Let

MY = multP (MX)P + RY ;

where log pair (Y;RY ) is movable.

Note, that Lemma III is based on A.Corti's Lemma, which handles log pair with two

normally intersected prime divisors and one movable boundary.

Observation. Divisors F and g�1(P ) intersect normally in S.

Therefore, I can apply A.Corti's Lemma to log pair

(V;MV ) = (V; (multO(RY ) +multP (MX)� 2)F +multP (MX)g�1(P ) +RV );

where RV = g�1(RY ). Last gives

multS(R2V ) � 4(3�multO(RY )�multP (MX))(1�multP (MX)):

Combining with

multO(R2Y ) � mult

2O(RY ) +multS(R

2V )

I get the following inequality.

Important Inequality I.

multO(R2Y ) � (multO(RY )� 1 +multP (MX))

2 + 8(1�multP (MX)):

91

Now I will bound multO(R2Y ) from above.

Let Z be a general enough hyperplane section of Y passing through point O.

Observation. Z is a smooth quintic surface in P3.

So,

(RY jZ)2 � multO(R

2Y ):

On the other hand,

(RY jZ)2 = ((Z �multP (MX)P )jZ)

2 = 5� 2multP (MX)� 3mult2P (MX):

Therefore, I obtained the following inequality.

Important Inequality II.

multO(R2Y ) � 5� 2multP (MX)� 3mult2P (MX):

Important Inequalities I and II imply

multP (MX) = 1 and multO(RY ) = 0:

Thus,

multO(MX) = multO(MY ) = multP (MX) +multO(RY ) = 1:

Therefore,

O 62 CS(X;MX):

92

x3. Sextic, septic and octic.

In this section I will consider application of results of Chapter I to an arbitrary smooth

hypersurfaces of degree N in PN in case when N = 6; 7 or 8.

Object. Let X be a smooth hypersurface of degree N in PN for N = 6; 7 or 8.

Note, that

Pic(X) = �ZKX and�KX � OPN(1)jX :

Main result of this section is the following theorem.

Theorem. X is birationally superrigid and could not be birationally transformed into

�brations on varieties with Kodaira dimension zero except �brations described in section 1

of this chapter.

Consider movable log pair (X;MX) with

KX +MX �Q 0:

From the methods of Chapters III and IV follows that Theorem is implied by the following

statement.

Central Theorem. Singularieties of (X;MX) are canonical, �(X;MX) = 0,

CS(X;MX) =

(;;

X \H for a linear subspace H � PN of dimension N � 2:

Moreover, if CS(X;MX) 6= ;, then

MX = ��1P (MP1);

where P is a pencil in j �KX j, such that CS(X;MX) = fBs(P)g:

From the results of section 1 of this chapter follows that Central Theorem is implied by

the following lemma.

Statement. CS(X;MX) contains no points.

Suppose CS(X;MX) contains point O. Consider log pair

(X;BX) = (X;HX +MX);

93

where

HX =N�4Xi=1

Hi

and divisors Hi are general hyperplane sections of X passing through point O.

Observation. (X;BX) is not log canonical in point O.

Let f :W ! X be a blow up of O and E = f�1(O). Then

a(X;BX ; E) = a(X;MX ; E)�N + 4

and proof of Lemma I in the previous section implies the following inequalities:

a(X;BX; E) > �1 and a(X;MX ; E) > 0:

Consider log pair

(W;BW ) = (W; (multO(MX)� 2)E + BW ) = (W; (multO(MX)� 2)E +HW +MW );

where HW = f�1(HX) and MW = f�1(MX). By construction

KW + BW �Q f�(KX +BX):

Proof of Lemma II of previous section implies that LCS(W;BW ) contains proper

irreducible subvariety S � E, such that variety S is not contained in HW and log pair

(W;BW ) is not log canonical in generic point of S.

Let S be an element of maximal dimension of LCS(W;BW ), such that S is a proper

subvariety of E, S 6� HW and log pair (W;BW ) is not log canonical in generic point of S.

Then proof of Lemma III of previous section implies that codimension of S is not 2 and

proof of Lemma IV of previous section implies that S is not a point.

Slip to 3-fold. Let Z = \N�4i=1 Hi.

Consider log pair

(Z;MZ) = (Z;MX jZ):

Due to generality of HX Z is smooth and log pair (Z;MZ) is movable.

94

Note, that adjunction formula implies

KZ +MZ �Q (KX +BX)jZ :

Which properties of log pair (Z;MZ) are inherited from (X;MX)?

Lemma. Log pair (Z;MZ) is not log canonical in point O.

Proof. Claim follows from Inverse of Adjunction (see section 3 of Chapter I). �

Let h : U ! Z be a blow up of point O and G = h�1(O). Then by construction

EjU = G; multO(MZ) = multO(MX) and h�1(MZ) =MW jU :

Consider log pair

(U;MU) = (U; (multO(MZ)� 2)G+MU );

where MU = h�1(MZ). Then adjunction formula implies

KU +MU �Q f�(KZ +MZ) �Q (KW + BW )jU :

Observation. Due to generality of HX

LCS(U;MU) = LCS(W; (multO(MX � 2)E +MW ) \ U:

Main Property of Z. LCS(U;MU) consists of one point and singularities of (U;MU)

are not log canonical in this point.

Proof. Last Observation implies that set LCS(U;MU) contains only points. V.V.Sho-

kurov's Relative Connectedness Theorem implies that set LCS(U;MU) contains exactly

one point and Lemma implies that log pair (U;MU) is not log canonical in this point. �

Note, that Main Property of Z implies that S is a linear subspace in E �= PN�2 of

dimension N � 4.

Let Y be a general enough hyperplane section of X passing through point O, such that

S � f�1(Y ):

95

Put

(Y;MY ) = (Y;MX jY ):

Warning. Y may be singular.

Nevertheless, log pair (Y;MY ) can be handled in the neighbourhood of point O, because

point O is smooth on Y .

Observation. Log pair (Y;MY ) is movable.

Let g : V ! Y be a blow up of O and F = g�1(O). Then by construction

S � F; EjV = F; multO(MY ) = multO(MX) and g�1(MY ) =MW jV :

Consider log pair

(V; (multO(MY )� 2)F +MV );

where MV = g�1(MY ).

Most Important Property of Y . Log pair

(V; (multO(MY )� 2)F +MV )

is not log canonical in generic point of S.

Proof. Restricting \everything" to the general hyperplane sections of X passing

through point O allows me to assume that W is 4-fold, V is 3-fold and S is a curve.

Now I can use arguments from the proof of similar statement in the previous section. �

Thus, A.Corti's Lemma (see section 3 of Chapter I) implies

multS(M2V ) > 4(3�multO(MY )):

So,

multO(M2Y ) � mult

2O(MY ) +multS(M

2V ) > mult2O(MY ) + 4(3�multO(MY )):

Hence,

N = (�KX)N�4 �M2

Y � multO(M2Y ) > (multO(MY )� 2)2 + 8:

96

SINGULAR FANO 3-FOLDS

x1. 3-fold with unique elliptic structure.

In this section I will apply results of Chapter I to singular double cover of P3.

Object. Let � : X ! P3 be a double cover rami�ed in sextic S, such that X has one

singular point O, which is a simple double point.

Note, that X is a Fano 3-fold with terminal Q -factorial singularities,

Pic(X) = Z and KX � ��(OP3(�1)):

Birational structure of X was studied in paper [Pu6], where birational superrigidity of

3-fold X was proved. However, except birational superrigidity X has other interesting

properties.

Elliptic Structure on X. Let f : W ! X be a blow up of singular point O. Then

linear system j �KW j is free and morphism

�j�KW j :W ! P2

is an elliptic �bration.

Purpose of this section is to prove the following result.

Main Theorem. X is not birationally isomorphic to other elliptic �brations.

Let (X;MX) be movable log pair on X with

KX +MX �Q 0:

It follows from the methods of Chapters III and IV that Main Theorem is implied by the

following statement.

Central Theorem. (X;MX) is canonical, �(X;MX) = 0 and

CS(X;MX) =

8>>>>>><>>>>>>:

fOg;

fBs(P)g; for pencil P in j �KX j; such that O 62 Bs(P);

fBs(P); Og; for pencil P in j �KX j; such that O 2 Bs(P);

;:

97

Moreover, if singularities of (X;MX) are not terminal, then MX = �1(MY ); where

: X 9 9 KY

is composition of � and projection from �(CS(X;MX)).

Note, that Criterion on Superrigidity (see section 2 of Chapter II) together with Central

Theorem implies that quintic X is birationally rigid.

Statement. To prove Central Theorem one should only prove canonicity of singularities

of log pair (X;MX) and describe locus CS(X;MX).

Proof. Suppose that log pair (X;MX) is canonical and

CS(X;MX) =

8>><>>:fOg;

fBs(P)g; for pencil P in j �KX j; such that O 62 Bs(P);

fBs(P); Og; for pencil P in j �KX j; such that O 2 Bs(P);

I will prove that

MX = �1(MY );

where : X 9 9 KY is composition of � and projection from �(CS(X;MX)).

Let Z be union of all curves in CS(X;MX) if CS(X;MX) contains a curve. Otherwise

let Z = O.

In case O 2 CS(X;MX) A. Corti's Theorem (see section 3 of Chapter I) implies

multO(MX) = 1:

Consider linear system H of surfaces in j � KX j containing Z. Choose birational

morphism f : W ! X, such that linear system f�1(H) is free, W is smooth and f is

an isomorphism outside of Z. Put

g = �H Æ f and (W;MW ) = (W; f�1(MX)):

Fix general enough divisor D in f�1(H).

There are four possible cases: �(Z) is not contained in S; �(Z) is a line in S not passing

through point �(O); �(Z) is a line in S passing through point �(O); Z = O.

98

Suppose that �(Z) 6� S. I may assume thatW contains exactly one f -exceptional divisor

lying over generic point of every irreducible component of Z. Then

MW jD �Q

kXi=1

ciFijD;

where all f(Fi) are points on X and all ci are rational. Therefore, MW lies in the �bers of

�bration g. Last implies the claim.

Suppose that �(Z) is a line in S not passing through the point �(O). I may assume

that morphism f is a composition of blow up of Z and blow up of a section of exceptional

surface of the �rst blow up. Then

MW jD �Q a(X;MX; E2)E2jD;

where E2 is an exceptional surface of the second blow up. On the other hand, E2jD is a

smooth rational curve on smooth K3 surface D. So,

a(X;MX; E2) = 0

and boundary MW lies in the �bers of �bration g. Last implies the claim.

Suppose that �(Z) is a line in S passing through point �(O). I may assume that f is a

composition of blow up of O, blow up of proper transform of Z and blow up of a section

of exceptional surface of the second blow up. Then

MW jD �Q (a(X;MX; E)E + a(X;MX ; E2)E2)jD;

where E and E2 is an exceptional surfaces of the �rst and third blow ups respectively. On

the other hand, EjD and E2jD are two smooth rational curves on smooth K3 surface D,

which are intersected transversally in one point. Thus,

a(X;MX ; E) = a(X;MX ; E2) = 0:

So, boundary MW lies in the �bers of �bration g. The latter implies the claim.

99

Suppose now Z = O. Then g is an elliptic �bration. I may assume that f is a blow up

of point O. For general enough �ber C of g

MW � C = 0:

Therefore, MW lies in the �bers of g. Last implies the claim. �

In order to prove Central Theorem consider four auxiliary lemmas.

Lemma I. Locus CS(X;MX) contains no smooth points of X.

Proof. Suppose that locus CS(X;MX) contains some smooth point P on X. Then

V.A.Iskovskikh & A.V.Pukhlikov's Theorem (see section 3 of Chapter I) implies

multP (M2X) = multP ((MX jHX)

2) � 4:

On the other hand,

2 = �KX �M2X � multP (M

2X):

�

Inequality

2 = �KX �M2X � 2multO(MX)

2

together with A. Corti's Theorem (see section 3 of Chapter I) imply the following result.

Lemma II. Log pair (X;MX) is canonical in O.

Thus, I may assume that CS(X;MX) contains some irreducible and reduced curve C.

Lemma III. �KX �C � 2 and �(C) is either a conic or a line.

Proof. Inequality

2 = �KX �M2X � multC(M

2X)(�KX) � C � �KX � C � deg(�(C))

implies the claim. �

Lemma IV. �(C) is a line.

Proof. Suppose that curve �(C) is not a line. Then inequality in the proof of Lemma

III implies that �KX �C = 2, �(C) is a conic, �jC is an isomorphism and

multC(MX) = 1:

100

Choose general enough divisor H in j �KX j. H is smooth K3 surface, which intersects

curve C exactly in two di�erent points x1 and x2.

Let g : V ! H be blow up of x1 and x2. Put E1 = g�1(x1) and E2 = g�1(x2). Then

linear system

jg�(HjH)� E1 � E2j

contains exactly one e�ective divisor D.

Note, that D is smooth curve of genus 2.

On the other hand,

(g�1(MX jH)) �Q g�(HjH)�E1 �E2

and

(g�1(MX jH))2 = 0:

Therefore, linear system jnDj does not have �xed components for n� 0 and D2 = 0.

For some n� 0 linear system jnDj is free and

�jnDj(V ) = P1:

Hence, for some integer k 2 (1; n] �bration �jnDj has multiple �ber kD. Last implies that

the genus of curve D is equal to 1. �

Proof of Central Theorem. Suppose

CS(X;MX) 6= ; and CS(X;MX) 6= fOg:

Then Lemmas I-IV imply that CS(X;MX) contains a smooth rational curve C such that

equality �KX � C = 1 holds.

If C � S, then proof of Statement leads to canonicity of log pair (X;MX) and description

of CS(X;MX). Therefore, I can assume that C 6� S. Thus

��1(�(C)) = C [ C 0;

where C 0 is smooth rational curve such that �KX � C 0 = 1.

101

Consider pencil H of surfaces in j � KX j containing C and C 0. Choose birational

morphism f : W ! X, such that pencil f�1(H) is free, W is smooth and f is an

isomorphism outside of C and C 0. Put

(W;MW ) = (W; f�1(MX))

and �x general enough divisor D in pencil f�1(H).

There are two possible cases: curves C and C 0 pass through point O; curves C and C 0

do not pass through point O. In the �rst case I have to show that

C 0 2 CS(X;MX) and O 2 CS(X;MX):

In the last case I have to show that

C 0 2 CS(X;MX):

Suppose curves C and C 0 pass through point O. I may assume that f is a composition

of blow up of O, blow up of proper transform of C and blow up of proper transform of C 0.

Then

MW jD �Q (a(X;MX; E)E + a(X;MX ; E2)E2)jD;

where E and E2 is an exceptional surfaces of the �rst and third blow ups respectively. On

the other hand, EjD and E2jD are two smooth rational curves on smooth K3 surface D,

which are intersected transversally in one point. Thus,

a(X;MX ; E) = a(X;MX ; E2) = 0:

Suppose curves C and C 0 do not pass through point O. I may assume that f is a

composition of blow up of C and blow up of proper transform of C 0. Then

MW jD �Q a(X;MX; E2)E2jD;

where E2 is an exceptional surfaces of the second blow up. On the other hand, E2jD is a

smooth rational curve on smooth K3 surface D. Last implies

a(X;MX ; E2) = 0:

�

102

x2. Singular quartic 3-fold.

In this section I will apply results of Chapter I to singular quartic 3-fold.

Object. Let X be a quartic 3-fold in P4 having one singular point O, which is locally

isomorphic to simple double point.

Note, that X is a Fano 3-fold with terminal Q -factorial singularities,

Pic(X) = Z and KX � OP4(�1)jX :

Assumption on X. Suppose in the following that X is general enough in the following

sense: X contains exactly 24 di�erent lines passing through point O.

Main purpose of this section is to prove the following result.

Main Theorem. For point O and every line C on X passing through point O there

are birational maps

O : X 9 9 KXO and C : X 9 9 KXC

correspondingly, such that both XO and XC are Fano 3-folds with canonical singularities

and �K3XO

= 2 and �K3XC

= 12 . Vice versa, for any Fano 3-fold Y having canonical

singularities, which is birationally equivalent to X,

Y �=

8>><>>:X;

XO;

XC for some line C on X passing through point O:

Note, that A.V.Pukhlikov proved in [Pu2] that X is birationally rigid and constructed

25 non-biregular birational automorphisms �O and �C , where C runs among lines on X

passing through point O. Moreover, he proved that such birational automorphisms �O and

�C together with biregular automorphisms generate all group of birational automorphisms.

Remark-Observation. Involutions O Æ �O Æ �1O and C Æ �C Æ

�1C are biregular.

Now I will show how to construct birational map O. Let f : W ! X be a blow up of

point O.

Observation. Divisor �KW is nef and big. Moreover, �K3W = 2.

103

So, for some integer n� 0 linear system j � nKW j gives birational morphism

�j�nKW j :W ! XO:

Corollary. By construction XO is Fano 3-fold with terminal Gorenstein singularities

and �K3XO

= 2.

Let O = �j�nKW j Æ f�1.

Remark. 3-fold XO is double cover of P3 rami�ed in sextic with 24 simple double

points.

Fix line C on quartic X � P4 passing through simple double point O. Now I will

construct birational map C .

Blow up f : W ! X point O and put Q = f�1(O). Blow up g : V ! W proper

transform f�1(C) of line C on W and put

E = (f Æ g)�1(C) and QV = g�1(Q):

Observation. Linear system j �KV j is free and gives elliptic �bration

j�KV j : V ! P2

having QV as a section.

Note, that restriction of morphism j�KV j to QV is birational and contracts two smooth

rational curves. Denote them C1 and C2.

Now consider nef and big divisor (f Æ g)�(�KX) �KV . One may easyly check that it

has zero intersection only with curves C1 and C2. Last leads to the following result.

Corollary. There is a op � : V 9 9 KV in curves C1 and C2.

Restriction of � to the general element of linear system j �KV j is an isomorphism. So,

anticanonical linear system j �KV j is free.

Remark. �(QV ) �= P2 and its normal bundle is OP2(�2).

Therefore, there is birational morphism

g : V ! W

contracting �(QV ) to the quotient terminal singular point P of type 12 (1; 1; 1).

104

Base Locus and Degree. �K3W

= 12 and Bs(j �KW j) = P .

Proof. Freeness of linear system j �KV j implies

Bs(j �KW j) = P:

Equality �K3W

= 12follows from

0 = �K3V= (g�(�KW )�

1

2�(QV ))

3 = �K3W�

1

2:

�

In particular, divisor �KW is nef and big. Therefore, linear system j � nKW j is free for

some n� 0 and gives birational morphism

�j�nKWj : W ! XC ;

where 3-fold XC is normal.

Observation. XC is Fano 3-fold with canonical singularities and �K3XC

= 12 .

Let

C = �j�nKWj Æ g Æ � Æ g

�1 Æ f�1:

Now I will prove Main Theorem.

Assumption. There is a birational map

� : X 9 9 KY;

where Y is a Fano 3-fold with canonical singularities.

Choose such integer n� 0 that linear system j � nKY j is free and put

(Y;MY ) = (Y;1

nj � nKY j):

Consider log pairs (X;MX), (XO;MXO) and (XC ;MXC ), which are birationally equivalent

to log pair (Y;MY ).

Reduction. Up to the action of Bir(X) one of log pairs

(X;MX); (XO;MXO) or (XC ;MXC )

105

has semiterminal singularities and Q -rationally trivial log canonical divisor.

Methods of Chapters III and IV imply that Main Theorem follows from Reduction.

From paper [Pu2] follows that for some � 2 Bir(X) singularities of movable log pair

(X; �(MX)) are canonical. Thus, composing � with some � 2 Bir(X) we may assume that

singularities of log pair (X;MX) are canonical. Therefore,

KX +MX �Q 0:

Theorem. For log pair (X;MX)

CS(X;MX) =

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

;;

fOg;

fCg for line C � X; O 2 C;

fO;Cg for line C � X; O 2 C;

fCg for line C � X; O 62 C;

fO;X \Hg for 2-dimensional linear subspace H � P4; O 2 H;

fX \Hg for 2-dimensional linear subspace H � P4; O 62 H:

Proof. Assume that singularities of log pair (X;MX) is not terminal and

CS(X;MX) 6=

8>>>>>><>>>>>>:

fOg;

fCg for line C � X; O 2 C;

fO;Cg for line C � X; O 2 C;

fCg for line C � X; O 62 C;

I will prove that

CS(X;MX) =

(fO;X \Hg for 2-dimensional linear subspace H � P4; O 2 H;

fX \Hg for 2-dimensional linear subspace H � P4; O 62 H:

Suppose that set CS(X;MX) contains smooth point P . Let HP be general enough

hyperplane section of X passing through point P . Then

4 = HP �M2X � multP (M

2X)

106

and by V.A.Iskovskikh & A.V.Pukhlikov's Theorem (see section 3 of Chapter I)

multO(MX) = 2:

Now blow up f :W ! X point P . Q -rational equivalence

f�1(MX) �Q �KW

implies that linear system j � nKW j has no �xed components for n� 0.

Let S be inverse image of surface HP on W . Then linear system j � KW jS j contains

exactly one e�ective divisor D and linear system jnDj is free for n � 0. On the other

hand, D2 = 0 implies

�jnDj(S) = P1:

Therefore, for some k 2 (1; n] kD is a multiple �ber of �j�nDj. Thus, �j�nDj should be

elliptic �bration, but arithmetical genus of curve D is 2.

Thus, set CS(X;MX) does not contains smooth points of X.

Note, that if every element of CS(X;MX) disjoint from point O, then the claim follows

from the results of section 2 of Chapter IV. Therefore, I may assume that set CS(X;MX)

contains curve C passing through point O.

Note, that C 2 CS(X;MX) implies inequality

multC(MX) � 1:

So, for general hyperplane section H of X

4 = H �M2X � multC(M

2X)H � C � deg(C):

Suppose �rst that C is smooth curve, which is not contained in two-dimensional linear

subspace in P4. Let f : W ! X be blow up of point O, g : V ! W be blow up of

curve f�1(C), Q = f�1(O), E be exceptional divisor of g and movable log pair (V;MV )

be birationally equivalent to log pair (X;MX). Then

MV �Q (f Æ g)�(�KX)�multO(MX)g�(Q)�E:

107

Curve C is an intersection of quadrics. In particular, divisor

D = (f Æ g)�(�2KX)� g�(Q)�E

is nef. Therefore,

0 � D �M2V = 7� 3deg(C) + 2multO(MX)� 2mult2O(MX) � 7

1

2� 3deg(C):

Keeping in mind that deg(C) is either 3 or 4 we get contradiction.

Suppose now that C is a rational curve of degree 4 with one double point, which is

not contained in any two-dimensional linear subspace in P4. Then curve C is contained in

unique hyperplane section of quartic X.

Take general enough divisor hyperplane H section of X. Consider intersection

H \ C = fx1; x2; x3; x4g:

Blow up g : V ! H points xi and put Ei = g�1(xi) for i = 1; : : : ; 4.

By assumption linear system

jg�(HjH)�4Xi=1

Eij

has exactly one e�ective divisor D and D2 = 0. On the other hand,

g�1(MX jH) �Q g�(HjH)�

4Xi=1

Ei:

Therefore, linear system jnDj is free for some integer n� 0. Moreover,

�jnDj(V ) = P1:

Thus, for some k 2 (1; n] �bration �jnDj has multiple �ber kD. Therefore, �jnDj is an

elliptic �bration. However, explicit calculations show that D has arithmetical genus 4.

Therefore, C is contained in some two-dimensional linear subspace T in P4 and I can

assume that deg(C) = 1; 2 or 3.

108

Suppose that curve C is not a line. Let HT be pencil in linear system j�KX j containing

surfaces passing through X \ T and D be general enough divisor in pencil HT . Assume

for simplicity that

X \ T = D \ T = C [rXi=1

Ci;

where all Ci are irreducible and reduced curves on surface D. Then on surface D

(rXi=1

Ci) � Cj = (DjD � C) � Cj = deg(Cj)� C � Cj

and on surface T �= P2

deg(Cj)� C � Cj = deg(Cj)� deg(C)deg(Cj) < 0:

If Cj contains point O, then curve Cj meets curve C at least in deg(Cj) points di�erent

from point O. Hence, on surface D for all curves Cj

(rXi=1

Ci) �Cj < 0:

Last implies that intersection form of curves Ci on surface D is negative de�ne.

Divisor

MX jD �multC(MX)C �rXi=1

multCi(MX)Ci

is nef on surface D and Q -rationally equivalent to divisor

rXi=1

(1�multCi(MX))Ci:

Thus, on surface D for all curves Cj

rXi=1

(1�multCi(MX))Ci � Cj � 0:

Therefore, all multCi(MX) � 1 and all Ci are containded in CS(X;MX), which implies

the claim.

109

Thus, I may assume that C is a line. Moreover, I may assume that CS(X;MX) contains

only lines and may be point O.

In case when set CS(X;MX) contains two intersected lines all previous arguments are

still valid. Therefore, I may assume that set CS(X;MX) consists of disjoint lines and may

be singular point O. In particular, there is a line L belonging to the set CS(X;MX), such

that L does not pass through point O.

Consider blow up f : W ! X of line L and log pair (W;MW ), which is birationally

equivalent to log pair (X;MX). Anticanonical linear system j �KW j is free and morphism

�j�KW j :W ! P2

is an elliptic �bration.

Let Z be a general enough �ber of �j�KW j. Then

MW � Z = 0:

Thus, MW lies in the �bers of �j�KW j. Last together with C 2 CS(X;MX) easyly leads

to contradiction with the movability of log pair (W;MW ). �

Proof of Reduction. I may assume that singularities of log pair (X;MX) are worse

than canonical.

Let (XO;MXO) and (XC ;MXC ) be movable log pairs, which are birationally equivalent

to movable log pair (X;MX), where C runs through all lines on quartic 3-fold X passing

through point O. I have to prove that either (XO;MXO) has semiterminal singularities

and KXO +MXO �Q 0, or one of log pairs (XC ;MXC ) has semiterminal singularities and

equivalence KXC +MXC �Q 0 holds.

Application of Theorem to log pair (X;MX) gives us six cases:

CS(X;MX) =

8>>>>>>>>>>>><>>>>>>>>>>>>:

fOg;

fCg for line C � X; O 2 C;

fO;Cg for line C � X; O 2 C;

fCg for line C � X; O 62 C;

fO;X \Hg for 2-dimensional linear subspace H � P4; O 2 H;

fX \Hg for 2-dimensional linear subspace H � P4; O 62 H:

110

Let H be a linear system consisting of hyperplane sections of quartic X, which contain

all elements of set CS(X;MX), and commutative diagram

W

f . & g

X�H9 9 K �H(X)

resolves indeterminacy of rational map �H.

Suppose

CS(X;MX) =

(fO;X \Hg for 2-dimensional linear subspace H � P4; O 2 H;

fX \Hg for 2-dimensional linear subspace H � P4; O 62 H:

Then for general enough �ber D of morphism g

f�1(MX)jD �Q

kXi=1

ciFijD;

where all ci 2 Q and every divisor Fi contracted by birational morphism f to the point

on quartic X. Last implies f�1(MX) lies in the �bers of g, which is impossible due to the

choice of (X;MX).

Now suppose

CS(X;MX) =

(fO;Cg for line C � X; O 2 C;

fCg for line C � X; O 62 C;

Then for general enough �ber D of morphism g

f�1(MX) �D = 0

and we get contradiction with the choice of log pair (X;MX).

Thus, either CS(X;MX) = fOg, or CS(X;MX) = fCg for some line C on X passing

through point O.

Suppose that CS(X;MX) = fCg for some line C on quartic X, which contains singular

point O. Blow up f : W ! X point O and then blow up g : V ! W strict transform of

line C on W .

111

Let QV = g�1(f�1(O)), E be exceptional divisor of g and log pair (V;MV ) be

birationally equivalent to log pair (X;MX). Then

KV +MV �Q (f Æ g)�(KX +MX) + a(V;MV ; QV )QV

and E �= P1 � P1.

Suppose that singularities of log pair (V;MV ) are not terminal. Then set CS(V;MV )

contains such curve CV � E, that CV dominates line C. Last contradicts to

MV �D = a(V;MV ; QV ) 2 Q \ (0; 1);

where D is a �ber of morphism �j�KV j passing through some point of curve CV .

Therefore, singularities of (V;MV ) are terminal. From the construction of birational

map C follows that movable log pair (XC ;MXC ) has semiterminal singularities and

KXC +MXC �Q 0:

Suppose CS(X;MX) = fOg. Let f : W ! X be blow up of O, Q = f�1(O) and log

pair (W;MW ) be birationally equivalent to (X;MX). Then A. Corti's Theorem implies

KW +MW �Q f�(KX +MX):

Birational morphism from W to XO is crepant for log pair (W;MW ). So, as in the previos

case I have to show that log pair (W;MW ) has terminal singularities.

Suppose, set CS(W;MW ) has point OW 2 Q, which is not contained in strict transforms

of all lines on X passing through point O. Then

multOW (M2W ) � 4

by V.A. Iskovskikh & A.V. Pukhlikov's Theorem. Intersecting M2W with general surface

in linear system j �KW j passing through point OW we get contradiction.

Therefore, we may assume that set CS(W;MW ) contains point OW in divisor Q, such

that OW 2 f�1(C) for some line C on X passing through O.

112

Let g : V ! W be blow up of f�1(C), E be unique exceptional divisor of birational

morphism g and log pair (V;MV ) be birationally equivalent to movable log pair (W;MW ).

Then by Inverse of Adjunction (see section 3 of Chapter I)

LCS(E;MV jE) 6= ;:

On the other hand, in Pic(E) divisor MV jE has type

(multC(MX);multC(MX))

and multC(MX) < 1. Last contradicts Quadric Lemma (see section 3 Chapter I).

Therefore, set CS(W;MW ) has no points.

Suppose that set CS(W;MW ) contains a curve Z in Q. Intersecting MW with rulings

of Q we conclude that Z is also a ruling of quadric Q.

Let g : V ! W be a blow up of curve Z, H be general surface in pencil j � KV j and

movable log pair (V;MV ) be birationally equivalent to movable log pair (W;MW ). Then

surface H is a smooth K3 surface and

MV jH �Q HjH � Z +kXi=1

Ci;

where Z and Ci are rational curves on H, f Æ g(Ci) is a line on quartic X passing through

point O, curve g(Ci) has non-empty intersection with curve Z and the following equality

holds

Z � Ci = 1:

We put k = 0 in case when quartic X contains no such line C passing through point O,

that f�1(C) intersects curve Z. Note, that k � 2.

On the other hand,

MV jH = multZ(MV )Z +kXi=1

multCi(MV )Ci:

Last implies Z 2 CS(V;MV ) and for all curves Ci if any Ci 2 CS(V;MV ). Proceeding as

in the previous cases we get contradiction with the choice of movable log pair (X;MX). �

113

REGULARIZATION OF BIRATIONAL AUTOMORPHISMS

x1. Explicit regularization.

Object. Let X be an arbitrary 3-fold.

Consider the following notation.

Regularization. Subset S � Bir(X) is said to be regularized on variety V by birational

map : X 9 9 KV and is called regularization of S if

Æ S Æ �1 � Aut(V ):

Note, that every regularizable subset of Bir(X) regularizable subgroup in Bir(X).

Example. Let � be birational automorphism of variety X regularizable by birational

map . Then cyclic group

f�njn 2 Zg � Bir(X)

is regularized by as well.

It is natural to consider the following problem.

Regularization Problem. Describe regularizable subgroups of the group of birational

automorphism of X and their regularizations.

It is known that every �nite subgroup of group Bir(X) is regularizable. I will give an

explicit way of such regularization.

New Object. Let G be �nite subgroup of Bir(X).

Fix very ample divisor H on X together with non-negative rational number � and put

(X;H�X) = (X;

Xg2G

�g(jHj)):

Note, that log pair (X;H�X) is G-invariant.

Lemma on Kodaira Dimension. �(X;H�X) = dim(X) for �� 0.

114

Proof. Kodaira dimension of movable log pair is not decreasing function of the

coeÆcients of the boundary. Thus

�(X;H�X) � �(X;�jHj):

One the other hand, ampleness of H leads to

�(X;�jHj) = dim(X) for m� 0:

�

Due to Lemma on Kodaira Dimension for some � � 0 Log Minimal Model Program

and Log Abundance Theorem (see [KeMaMc]) imply the existence of birational map

� : X 9 9 KV;

such that log pair

(V;H�V ) = (V; �(H�

X))

is a canonical model of log pair (X;H�X). Then log pair (V;H�

V ) is � Æ G Æ ��1-invariant

and Uniqueness Theorem (see section 1 of Chapter I) implies that

� ÆG Æ ��1 � Aut(V ):

Observation. Given construction is an explicit way of regularization of group G.

Second Main Theorem of paper [Sh2] implies that for some rational number Æ > � ca-

nonical model of log pair (X;HÆX) is log pair (V;HÆ

V ). In particular, both log canonical

divisors

KV +H�V and KV +HÆ

V

are Q -Cartier divisors. Therefore, canonical divisor

KV �Q (KV +H�V )�

�

Æ � �[(KV +HÆ

V )� (KV +H�V )]

is Q -Cartier too. The latter implies that singularities of 3-fold V are canonical. So, I can

apply � ÆG Æ ��1-invariant Minimal Model Program to V .

115

Corollary. Group G � Bir(X) can be explicitly regularized by such birational map

: X 9 9 KY;

that 3-fold Y has canonical singularities and either KY is nef or Y possess the structure

of ÆG Æ �1-invariant Fano �bration.

Note, that I could consider � Æ G Æ ��1-invariant resolution of singularities of V and

repeat previous arguments.

Remark. Group G � Bir(X) can be regularized by birational map : X 9 9 KY , such

that singularities of 3-fold Y are terminal Q -factorial and either KY is nef or Y possess

the structure of ÆG Æ �1-invariant Fano �bration.

116

x2. One example.

In this section I will consider one example of explicit regularization.

Object. Fix double cover

� : X ! Q � P4

of a smooth 3-fold quadric Q rami�ed in smooth octic S � Q.

Note, that X is a smooth Fano 3-fold with Pic(X) �= Z.

Lines on X. Curve C � X is called a line on X if �KX � C = 1.

Note, that X has one-dimensional family of lines. Moreover, every such line C induces

involution �C in Bir(X), such that involution �C is not biregular when line �(C) is not

contained in octic S (see [Is3]).

Remark. V.A.Iskovskikh proved in [Is3] that birational involutions �C and biregular

automorphisms of 3-fold X generate all group Bir(X).

Due to construction in the previous section involutions �C are regularizable on some

Fano �bration with canonical singularities. On the other hand, 3-fold X is birationally

rigid due to [Is3].

Corollary. For a given line C on X there is a birational map : X 9 9 KXC, such that

variety XC is Fano variety with canonical singularities and

Æ �C Æ �1 � Aut(XC):

Note, that Fano 3-folds with canonical singularities birationally equivalent to X were

classi�ed in section 4 of Chapter 4. Namely, for every line C on 3-fold X there is a birational

non-biregular map

C : X 9 9 KXC ;

such that XC is a Fano 3-fold with canonical singularities and �K3XC

= 12 .

One can naturally expect that

= C and XC �= XC :

I will give an explicit construction of birational map , which will imply that = C .

117

Line. Fix some line C on 3-fold X, such that �(C) 6� S.

For some rational � > 0 consider movable log pair

(X;H�X) = (X;�j �KX j+ ��C(j �KX j)):

Then explicit calculations show that singularities of movable log pair (X;H�X) are terminal

for � < 110 , canonical for � = 1

10 and not canonical for � > 110 . Moreover,

�(X;H�X) =

8>>>>><>>>>>:

�1 for � <1

10;

0 for � =1

10;

3 for � >1

10:

So, for � > 110 Uniqueness Theorem (see section 1 of Chapter I) implies that birational

map I(X;H�X) is a regularization of birational involution �C .

Consider blow up f :W ! X of line C and denote Z unique base curve of anticanonical

linear system j �KW j. Then blow up g : V ! W curve Z, put G = g�1(Z) and consider

movable log pair

(V;H�V ) = (V; (f Æ g)�1(H�

X)):

Observation. Singularities of movable log pair (V;H�V ) is terminal for all � > 0.

Thus, I can apply Log Minimal Model Program to log pair (V;H�V ). For � > 1

10 the

latter gives birational map � : V 9 9 KY , such that movable log pair

(Y;H�Y ) = (Y; �(H�

V ))

has terminal Q -factorial singularitiess and divisor KY +H�Y is nef and big.

Remark. I proved in section 1 of Chapter IV that birational map � is a composition of

op in some curve T contained in divisor G and contraction of proper transform of G into

cyclic quotient singular point of type 12 (1; 1; 1).

In case � = 110 birational morphism f :W ! X is crepant for to log pair (X;H�

X) and

KY +H�Y �Q 0:

118

Therefore, anticanonical divisor �KY is nef and big.

Now I can apply Log Abundance Theorem (see [KeMaMc]) to log pair (Y;H�Y ) for � >

110

and get birational morphism

I(Y;H�Y ) = j�n(KY +H�

Y)j for n� 0:

By construction

XC = I(Y;H�Y )(Y ):

Note, that Double Quadric Theorem (see section 4 of Chapter IV) implies the following.

Finite Subgroups of Bir(X). Every �nite subgroup of group Bir(X) is either subgroup

of Aut(X) or subgroup of Aut(XC) for some line C on X.

For every line C on X group Aut(XC) canonically contains group Z2 that induces

birational involution �C of 3-fold X. Moreover, one can show that for general enough X

Aut(X) �= Aut(XC) �= Z2:

Corollary. For general double quadric X all �nite subgroups of Bir(X) are isomorphic

to Z2 and coincide with conjugate classes of groups generated by involutions �C or involution

induced by double cover �.

Birational rigidity of 3-fold X together with �nal Remark in the previous section imply

that every �nite subgroup in Bir(X) can be can be regularized on some Fano 3-fold with

terminal Q -factorial singularities. On the other hand, singularities of 3-folds XC are not

Q -factorial by construction above.

Interesting Corollary. Every �nite subgroup in Bir(X) are conjugated to some �nite

subgroup in Aut(X).

119

x3. Question of Yu.I.Manin.

In this section I will answer one question of Yu.I.Manin.

Object. Let X be a smooth cubic surface in P3 with Pic(X) = Z over �eld F.

Note, that condition Pic(X) = Z implies that �eld F is not algebraically closed.

Group. Let G be a group of birational automorphisms of cubic X generated by Bertini

and Geizer involutions (see section 1 of Chapter III).

Note, that construction of Bertini and Geizer involutions imply that G is normal

subgroup in Bir(X).

Yu.I.Manin's Theorem. Cubic X is birationally rigid and group Bir(X) is semidirect

product of G and Aut(X).

Question. Which elements of Bir(X) has �nite order?

Note, that elements conjugated to biregular automorphisms, Bertini or Geizer involu-

tions has �nite order.

Remark. Let � be biregular automorphism of cubic X and � be either Bertini or Geizer

involution of X, such that � Æ � = � Æ �. Then � Æ � has �nite order.

D.S.Kanevsky described in [Ka] elements ofG, which has �nite order. Note, that method

of [Ka] are purely group-theoretic and based on relations in group G described in [Ma2].

Birational Automorphism. Let � be an element of �nite order in Bir(X).

In the rest of this section I will the following result.

Answer. � is conjugated to one of the following birational automorphisms: biregular

automorphism; Bertini involution; Geizer involution; composition of commuting biregular

automorphism and either Bertini or Geizer involution.

Note, that Answer and Yu.I.Manin's Theorem implies the following.

Corollary. All elements of the �nite order in group G are conjugated either to Bertini

or Geizer involutions.

Birational rigidity of X and �nal Remark in the section 1 of this chapter imply that

birational automorphism � can be regularized on some smooth del Pezzo surface Y . Thus,

there is birational : X 9 9 KY , such that

Æ � Æ �1 2 Aut(Y ):

120

On the other hand, Cubic Theorem (see section 4 of Chapter III ) implies the following.

Statement. Y is biregular either to X or to blow up of X.

Therefore, I may assume that there is a blow down f : Y ! X and K2Y = 1 or 2.

Suppose that K2Y = 2. Then

Pic(Y ) = ZKY � ZE;

where E is an exceptional divisor of f .

Note, that action of Æ � Æ �1 on Pic(Y ) should preserve KY . Moreover, if

Æ � Æ �1(E) � E;

then f Æ Æ � Æ �1 Æ f�1 is biregular. Therefore, I can assume that Æ � Æ �1 does not

preserve curve E. Then explicit calculations implies

( Æ � Æ �1)�(KX) � 2f�(KX) + 3E and ( Æ � Æ �1)�(E) � �f�(KX)� 2E:

From the construction of Geizer involution (see section 1 of Chapter III) follows that

for some Geizer involution � 2 Bir(X) action of

Æ � Æ �1 Æ f�1 Æ � Æ f

on Pic(Y ) is trivial. So, f Æ Æ � Æ �1 Æ f�1 Æ � is biregular automorphism of cubic X

preserving point f(E). The latter implies that f Æ Æ � Æ �1 Æ f�1 Æ � commutes with �.

Suppose now that K2Y = 1 and

Pic(Y ) = ZKY � ZE;

where E is an exceptional divisor of birational morphism f .

I can assume that the action of Æ � Æ �1 on Pic(Y ) is not trivial, because otherwise

birational automorphism f Æ Æ � Æ �1 Æ f�1 is biregular. So, one can calculate that

( Æ � Æ �1)�(KX) � 5f�(KX) + 6E and ( Æ � Æ �1)�(E) � �4f�(KX)� 5E:

121

From the construction of Bertini involution (see section 1 of Chapter III) follows that

for some Bertini involution � 2 Bir(X) composition f Æ Æ � Æ �1 Æ f�1 Æ � is biregular

automorphism of cubic X preserving f(E). Therefore, f Æ Æ � Æ �1 Æ f�1 Æ � commutes

with �.

Now I will consider remaining case: K2Y = 1 and

Pic(Y ) = ZKY � ZE1 � ZE2;

where E1 and E2 are exceptional curves of morphism f .

As in the previous two cases I can assume that the action of Æ � Æ �1 on Pic(Y ) is

not trivial. Moreover, same arguments let me to assume that

Æ � Æ �1(E1 [ E2) 6= E1 [E2:

Suppose, that curve E1 is Æ � Æ �1-invariant. Let g : Y ! V be a contraction of

curve E1. Then g Æ Æ � Æ �1 Æ g�1 is biregular automorphism of V . On the other hand,

surface V is a smooth del Pezzo surface with K2V = 2 and from the case K2

Y = 2 follows

that birational automorphism f Æ Æ � Æ �1 Æ f�1 is a composition of a Geizer involution

of X with some commuting biregular automorphism of cubic X.

I can assume that neither E1 nor E2 is Æ � Æ �1-invariant. Then

( Æ � Æ �1)�(KX) � 5f�(KX) + 6E1 + 6E2;

( Æ� Æ �1)�(E1) � �2f�(KX)�2E1�3E2 and ( Æ� Æ

�1)�(E2) � �2f�(KX)�3E1�2E2:

Construction of Bertini involution implies that f Æ Æ � Æ �1 Æ f�1 is a composition of

commuting Bertini involution of cubic X and biregular automorphism.

122

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VITAE

IVAN CHELTSOV

EDUCATION

Æ M.S. in Mathematics, Johns Hopkins University, 1997

Æ M.S. in Mathematics, Steklov Institute of Mathematics, 1996

Æ B.A. in Mathematics, Moscow State University, 1995

PERSONAL

Æ Born: October 10, 1973 (Moscow, Russia)

Æ Citizenship: Russian

Æ Languages: Russian (native), English ( uent)

Æ Hobbies: Books, Cinema, Theatre, Travel

Æ Home Page: http://www.math.jhu.edu/~cheltsov

Æ E-mail: cheltsov@yahoo.com

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