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AksoylKhamsi: Nonstandard Methods in Fixed Point Theory Andersson: Topics in Complex Analysis Aupetit: A Primer on Spectral Theory BachmanlNarici!B«kenstein: Fourier and Wavelet Analysis Biidescu: Algebraic Surfaces BalakrishnanlRanganathan: A Textbook of Graph Theory Balser: Formal Power Series and Linear Systems of Meromorphic Ordinary

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(continued after index)

Kenji Matsuki

Introduction to the Mori Program

With 61 figures

, Springer

Kenji Matsuki Department of Mathematics 1395 Mathematical Science Building Purdue University West Lafayette, IN 47907-1395 USA

Editorial Board ( North America):

S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA

K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA

F. W. Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109-1109 USA

Mathematics Subject Classification (2000): 15-01, 15A04, 15A69, 16-01

Library of Congress Cataloging-in-Publication Data Matsuki, Kenji, 1958-

Introduction to the Mori Program 1 Kenji Matsuki. p. cm. - (Universitext)

Includes bibliographical references and index. ISBN 978-1-4419-3125-2 ISBN 978-1-4757-5602-9 (eBook) DOI 10.1007/978-1-4757-5602-9 l. Algebraic varieties--Classification theory. I. Ti tie.

QA564 .M38 2001 516.3'53-dc21 00-067917

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© 2002 Springer Science+ Business MediaN ew York Originally published by Springer-Verlag New York, lnc. in 2002 Softcover reprint of the hardcover 1 st edition 2002

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ISBN 978-1-4419-3125-2 SPIN 10659699

Preface

This book started as a collection of personal notes that I made to help me to un-derstand what we call the Mori program, a program that emerged in the last two decades as an effective approach toward the biregular and/or birational classifica-tion theory of higher-dimensional algebraic varieties. In some literatures the Mori program restrictively refers to an algorithm, called the minimal model program, to produce minimal models of higher-dimensional algebraic varieties. (Classically, the construction of minimal models was known only for algebraic varieties of di-mension less than or equal to 2.) Here in this book, however, we use the word in a broader sense to represent a unifying scheme that is a fusion of the minimal model program and the so-called Iitaka program.

As such, I had no hesitation to, or rather even made elaborate efforts to, extract nice arguments from the existing literature whenever it seemed appropriate to fit them into a comprehensible development of the theory, even to the extent of copying them literally word by word. I am particularly aware of the original sources of the subject matters in the following list:

Chapter 1.

Barth-Peters-Van de Ven [I], Beauville [1], Clemens-Kollar-Mori [I], Griffiths-Harris [1], Hartshorne [3], litaka [5], Kawamata-Matsuda-Matsuki [1], Kodaira [2][3][4][5], Kollar [5], Mori [2][3], Reid [9], Shafarevich [I], Wilson [I]

Chapter 2.

Iitaka [2] [3] [5], Kawamata [1][2], Vojta [1]

VI Preface

Chapter 3.

C1emens-Kollar-Mori [1], Corti [1], Corti-Pukh1ikov-Reid [1], Iitaka [1] [4][5], Kawamata [4][5][6][12], Kawamata-Matsuda-Matsuki [1], Kollar [5], Kollar et al [1], Miyaoka-Mori [1], Mori [2][3][4][5], Reid [2][3][6][7][8], Sarkisov [3], Shokurov [1][2]

Chapter 4.

Artin [1][2][5], Brieskom [1], Kawamata [2][9], Mori [4], Mumford [3], Reid [2][3][7]

Chapter 5.

Clemens-Kolhir-Mori [1], Esnault-Viehweg [I], Kawamata [3]. Kodaira [1], Kollar [3][4][7], Viehweg [2]

Chapter 6.

Angehm-Siu [I], Ein-Lazarsfeld [I], Kawamata [4][6], Kawamata-Matsuda-Matsuki [I], Shokurov [1]

Chapter 7.

Clemens-Kolhir-Mori [I], Kawamata [5], Kollar [1], Mori [2]

Chapter 8.

Kawamata [5], Kawamata-Matsuda-Matsuki [I). Mori [2]

Chapter 9.

Kawamata [9], Kawamata-Matsuda-Matsuki [I], Mori [5], Shokurov [I]

Chapter 10.

Kawamata [5)[13] Kollar [I] [12] Miyaoka-Mori [1] Mori [I] [2]

Chapter 11.

Iitaka [2][3][5], Kawamata-Matsuda-Matsuki [I] Shokurov [2]

Chapter 12.

Kawamata [9], Kollar [8], Reid [3]

Chapter 13.

Corti [1], Reid [6], Sarkisov [3], Takahashi [1][2][3]

Chapter 14.

Danilov [1][2], Fulton [2], Oda [1][2], Reid [5]

I learned about the Mori program from my teachers, S. Iitaka, Y. Kawamata, J. Kollar, S. Mori, M. Reid, and V. V. Shokurov. My indebtedness to them goes far beyond what I can express in words both mathematically and personally.

Preface vii

My personal notes grew into the present form in the process of communicat-ing what I learned about the Mori program through seminars and classes held at Brandeis University and Purdue University. My thanks are due to D. Arapura, J. Lipman, T. Matsusaka, and the students who clarified many ambiguities and mistakes in my presentation of this beautiful theory. Special thanks go to D. Eisen-bud, without whose encouragement this book would have never converged into the present form.

I claim no originality except for simplification or reworking of some classical results (e.g., the analysis of rational double points as canonical singularities and the description of log terminal singularities as hyperquotients in Chapter 4) and for some explicit presentation of folklore-type results that are well known to experts but have never appeared in the literature (e.g., the toric Sarkisov program in Chapter 14). My priority is to present the Mori program in as easy and digested a form as possible, with the degree of motivational background that many have wished for but has not hitherto been revealed, and I apologize to those researchers who feel that the beauty of their original ideas has been deformed or lost in the process. Prerequisites are such that a graduate student who has read Hartshorne [3] or Iitaka [5] should have no difficulty understanding the material. (That is how much background knowledge I had when I started learning the subject.) My preference leans toward the geometric side rather than the purely algebraic, and this has made me feel free to switch back and forth between the analytic category and algebraic category.

This book consists of what I feel should roughly be the basics for an understand-ing of the global picture of the Mori program, leaving the hard-core analysis of 3-dimensional terminal singularities and the proof of the existence of flips to the original research literature. But this does not mean that the reader has to read the entire 400 plus pages to get a rough idea of what the Mori program is all about. The introduction in Chapter 0 and the overview in Chapter 3 should provide an easy guide for the remaining chapters, each of which could be read separately depending on the interest of the reader.

After circulating the first draft of this book in 1997, I received detailed sugges-tions and corrections from several people after some very careful readings, despite the numerous mistakes contained in the draft. Thanks are due to the attentions of D. Abramovich, V. Alexeev, B. Hassett, K. Hunt, S. Kovacs and V. Masek. Especially the notes of K. Oguiso and J. Sakurai from the seminars held at Tokyo University were extremely helpful in revising the book.

A stay at Warwick University was crucial for the book at the final stage of its preparation. M. Reid provided linguistic, mathematical, and philosophical help to the author with warm hospitality. S. Mukai gave private tutoring sessions to the author on several topics through stimulating conversations in the dark winter nights of England.

I would like to thank Ina Lindemann of Springer-Verlag, who kindly and pa-tiently waited for the completion of the manuscript, which was prepared and typeset using AvtST(3X.

I would also like to thank David Kramer, the copyeditor, who made an enormous effort to make this book readable.

viii Preface

The calligraphy on the front cover is done by Yumiko Takahashi. This book was conceived and written during a most turbulent and happiest time,

when our first baby, Mark Takamichi Matsuki, was born. I would like to dedicate this book to my wife, Dina Matsuki, known as Dinochka, who brought me a cup of tea every night after attending to crying Mark.

Contents

Preface

List of Notation

Introduction: The Tale of the Mori Program

1 Birational Geometry of Surfaces l.l Castelnuovo's Contractibility Criterion ...... . 1.2 Surfaces Whose Canonical Bundles Are Not Nef I . 1.3 Surfaces Whose Canonical Bundles Are Not Nef II 1.4 Basic Properties of Mori Fiber Spaces in Dimension 2 1.5 Basic Properties of Minimal Models in Dimension 2 . 1.6 Basic Properties of Canonical Models in Dimension 2 1.7 The Enriques Classification of Surfaces. 1.8 Birational Relation Among Surfaces

v

xiii

1

9 10 21 36 42 51 75 83 88

2 Logarithmic Category 109 2.1 Iitaka's Philosophy. . . . . . . . . . 110 2.2 Log Birational Geometry of Surfaces 118

3 Overview of the Mori Program 129 3.1 Minimal Model Program in Dimension 3 or Higher 131 3.2 Basic Properties of Mori Fiber Spaces in Dimension 3 or

Higher . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.3 Basic Properties of Minimal Models in Dimension 3 or Higher. 153 3.4 Birational Relations Among Minimal Models and Mori Fiber

Spaces in Dimension 3 or Higher . . . . . . . . . . . . . . .. 155

x Contents

3.5 Variations of the Mori Program 161

4 Singularities 163 4.1 Terminal Singularities 164 4.2 Canonical Singularities 168 4.3 Logarithmic Variations 173 4.4 Discrepancy and Singularities 179 4.5 Canonical Cover . . . . . . . 183 4.6 Classification in Dimension 2 191

5 Vanishing Theorems 250 5.1 Kodaira Vanishing Theorem . . . . . . . 251 5.2 Kawamata-Viehweg Vanishing Theorem 259

6 Base Point Freeness of Adjoint Linear Systems 268 6.1 Relevance of Log Category to Base Point Freeness of Adjoint

Linear Systems. . . . . . . . . . . . 269 6.2 Base Point Freeness Theorem . . . . 272 6.3 Nonvanishing Theorem of Shokurov 279

7 Cone Theorem 286 7.1 Rationality Theorem and Boundedness of the Denominator 286 7.2 Cone Theorem . . . . . . . . . . . . . . . . . . . . . . . . 297

8 Contraction Theorem 301 8.1 Contraction Theorem ..... 30 I 8.2 Contractions of Extremal Rays 306 8.3 Examples............ 309

9 Flip 9.1 9.2

Existence of Flip . . Termination of Flips

10 Cone Theorem Revisited 10.1 Mori's Bend and Break Technique 10.2 A Proof in the Smooth Case After Mori . 10.3 Lengths of Extremal Rays

317 318 322

324 325 330 332

11 Logarithmic Mori Program 338 11.1 Log Minimal Model Program in Dimension 3 or Higher 338 11.2 Log Minimal Models and Log Mori Fiber Spaces in Dimension

3 or Higher. . . . . . . . . . . . . . . . . . . . . . . . . . 342 11.3 Birational Relations Among Log Minimal Models and Log

Mori Fiber Spaces . . . . . . . . . . . . . . . . . . . . . . 345

Contents Xl

12 Birational Relation among Minimal Models 349 12.1 Flops Among Minimal Models . . . . . . . . . . . . . 350 12.2 Chamber Structure of Ample Cones of Minimal Models 358 12.3 The Number of Minimal Models Is Finite (?!) .. . . . 363

13 Birational Relation Among Mori Fiber Spaces 368 13.1 Sarkisov Program . . . . . . . . . . 369 13.2 Termination of the Sarkisov Program 392 13.3 Applications............ 399

14 Birational Geometry of Toric Varieties 413 14.1 Cone Theorem and Contraction Theorem for Torlc Varieties 413 14.2 Torlc Extremal Contractions and Flips ... 14.3 Toric Canonical and Log Canonical Divisors 14.4 Torle Minimal Model Program 14.5 Torlc Sarkisov Program .......... .

References

Index

425 442 447 449

455

469

List of Notation

C the field of complex numbers ]p'" the projective space of dimension n

MPI '" 4 the four major strategic schemes of the Mori program I C a nonsingular projective curve 2

Kc the canonical divisor on C 2

nZ' the sheaf of regular holomorphic 2-forms 2 ]p'1 the projective space of dimension I 2

g(C) = hO( C, Od K c)) the geometric genus of a curve C 2 degKc the degree of the canonical divisor K c 2 Aut(C) the group of automorphisms of C 2

PGLI(C) the projective general linear group of dimension lover C 2 Al the affine space of dimension I 2

MMP the Minimal Model Program 2 C2 the self-intersection number of a curve C 3

Ks'C the intersection numer of the canonical divisor with a curve C 3 Xmin a minimal model obtained from X via MMP 5 Xmori a Mori fiber space obtained from X via MMP 5

K the Kodaira dimension 5 Z the locally constant sheaf with coefficient in the ring of

integers Z 8 Ox the sheaf of holomorphic functions 8 0* x the sheaf of invertible holomorphic functions 8

Hi(X, Z) the ith sheaf cohomology group with respect to the usual topology, is isomorphic to the ith singular cohomology group 8

BlppT the blowup of a point p on a nonsingular surface T 11 mp the maximal ideal of a point p 11

XIV List of Notation

Projffid,,:om~ the projective scheme of a graded Or-algebra E9d,,:om~ 11 Pic(S) the Picard group of S 11

J1,* the pullback of a line bundle or a Cartier divisor by a morphism J1, 11

dimIRH2(T, JR.) the dimension of H2(T, JR.) over JR. 11 JR. the field of real numbers 11

X(Os(D) the Euler characteristic of a sheaf Os(D) 11 hi(E, OE) = dimcHi(E, OE) 12

n*0i; the direct image sheaf of a i; by n 12 Supp(Coker) the support of a sheaf Coker 13

1J1,* HI the complete linear system associated to a divisor J1, * H 13 Ill' HI the morphism associated to a linear system 1J1,* HI 13

J1,*E the image of E by a morphism J1, as a cycle 14 i»O for i sufficiently large 17 NE/ X the nonnal bundle of E in X 20

H 2(S, JR.) the 2nd (singular) homology group of S with coefficients in JR. 22

[D] the class of a curve Din H2(S, JR.) 22 Q>o the set of positive rational numbers 22

ZI(S) the group of I-cycles with coefficients in Z 23 degcJ1,* L the degree of a line bundle J1, * L on a curve C 23

numerical equivalence 23 NI(S) = {Pic(S)/ =} ® JR. 23 NI(S) = {ZI(S)/ =} ® JR. 23

peS) = dimIRNI(S) = dimIRNI(S), the Picard number 23 NE(S) the convex cone generated by effective I-cycles 23 NE(S) the closure of NE(X) in NI (S) 23

HI,I(S, JR.) = H 1,1 (S, C) n H 2(S, IR), the space of the elements of the 2nd cohomology group represented by the real harmonic forms of type (I, I) 24

cl(L) the first Chern class of a line bundle L 24 LJ.. = {z E NI (S); L . z = O} 25 sup the supremum 26

N the set of natural numbers (strictly positive integers) 26 SuppeD + aF) the support of a divisor D + a F 27

dim V the dimension of V as an algebraic variety 27 min the minimum 28

torsion the torsion subgroup 33 Spec leLl.OS the spectrum of an Ow,-algebra leL I, as 34

the equivalence relation among points 34 Re an extremal ray 36

JR.+ = JR.,,:o the set of nonnegative real numbers 36 [C] the numerical class of a curve C in N) (S) 36

IR+ [C] = JR.,,:o[C] the half-line generated by the class [C] in NI(S) 36

NE(S)Ks?:O contRI

K(S) ¢-l(p)

¢-l(P)red

deg IHI bi(S) e(S)

HP,q(S)

hp,q(S)

V* transdeg K(S, D)

A~B

List of Notation xv

= {z E NE(S); Ks . Z :::: O} the contraction of an extremal ray R/ the Kodaira dimension of S the scheme-theoretic fiber of P the fiber ¢-l(p) with the reduced structure the degree of a finite morphism IHI = dimcHi(S, C) the ith Betti number of S = I:( -IY bi(S) the topological Euler characteristic of S = c2(Ts ) the second Chern class of S the space of the elements of the (p + q )th cohomology group represented by the harmonic forms of type (p, q) the complex conjugate of Hp,q(S) = dimcHP,q(S)

the dual of the vector space V the transcendental degree the Iitaka dimension of the divisor D on S the linear equivalence of divisors A approximates B the canonical model of S the locally constant sheaf with coefficients in C

37 40 43 44 44 48 48

48 48

48 49 49 51 52 52 54 55 55

over Fp 63 Pico(F) the group of line bundles of degree 0 on F 65 G U G' the disjoint union of G and G' 69

(Rlcx*Os)V the dual of the line bundle Rlcx*Os 71 W Fi the dualizing sheaf for Fi 72

WSjC the relative dualizing sheaf of S over C 75 Proj R the projective scheme of a graded ring R 76

An, Dn, £6, £7, £-r, the possible types of the dual graphs of contractible (-2)-curves 77

An, Dn, £6, £7, £g the semidefinite dual graphs of noncontractible (-2)-curves 78

(Ii) ) the matrix with (i, j)-entry Ii) 79 Pg(S) = hOeS, Os(Ks» the geometric genus of S 84 q(S) = hl(S, Os) the irregularity of S 84

£( K s ) the total space of the canonical bundle K s 86 codimT(T - U) the codimension of T - U inside of T 90

div(f* x) the divisor associated to the (rational) function 1* x 91 J)E i the valuation associated to a divisor £i 91

¢ : S 1 - -~S2 a birational map ¢ from S I to S2 92

--~

birat a birational map 95

xvi List of Notation

IF, (JL, A, e)

JL =w

A n!

ho~aloidall H' I

multpi 1-£ e

(X, D) DiffxD

R10g n1(log D)

k(X)

n~(log D) K

K*

OK O~

MWgrp (5, B)min

(51, B)mori

Bv

6 A/(i)

qx, y, z](i) ~

Exc(¢)

the one-point blowup of ]P'2 the Sarkisov degree the quasi-effective threshold the numerical equivalence over W the maximal multiplicity = n . (n - 1) . (n - 2)· . ·3 ·2· 1 the linear system consisting of the homaloidal transforms a*a'*1-£' for 1-£' E I H' I the multiplicity of a divisor 1-£ at the point Pi the number of the crepant exceptional divisors a logarithmic pair the supplementary term for the generalized addjunction K x + DID = K D + Diff x D to hold the log ramification divisor the sheaf of logarithmic I-forms along D the field of rational functions of X over the base field k = I\m n1(log D) a number field = K - {a} the ring of algebraic integers in the number field K = OK - {a} the Mordell-Weil group a log minimal model obtained from (5, B) via Log MMP a log Mori fiber space obtained from (5, B) via Log MMP the category of log pairs (5, B) consisting of nonsingular surfaces 5 and boundary Q-divisors B with only normal crossings the category of log pairs (5, B) consisting of normal projective surfaces 5 and boundary Q-divisors B with only log terminal singularities = 1*-1 B + 1: E j, the boundary divisor associated to a resolution 1: V --+ 5 of a log pair (5, B) the strict transform of the divisor B the category of log pairs (5, B) consisting of normal projective surfaces 5 and boundary Q-divisors D with only Q-factorial and log terminal singularities the category of nonsingular projective varieties the quotient of the abelian variety A by the involution i the ring of invariants in C[x, y, z] by the involution i the category of normal projective variety with only Q-factorial and terminal singularities the exceptional locus for a birational morphism ¢

95 96 97 97 97 97

97 98 98

109

109 110 112 112 112 112 112 113 113 113

118

118

119

120

121 121

122 131 132 132

133 134

List of Notation xvii

the map induced by the cycle map i*:ZI(Ei ) ---+ ZI(X) 136

¢:X+ ---+ Y the flip of a flipping contraction ¢:X ---+ Y 137 Div(X) the group of Cartier divisors on X 137

X --~ Y X ]P'I X is birational to Y x ]P'I 143 birational

SymmF the mth symmetric power of a sheaf F 143 C[Y]m the mth graded part HO(y, CJy(mH» ofagradedring

qy] = EBm>oHo(y, Oy(mH» 144 p(p3 x Y j Y) the relative Picard number of]P'3 x Y over Y 147

C(t) the function field of]P'~ 148 k(P) the residue field at a point P 149

[k(P):C(t)] the extension degree of k(P) over C(t) 149 C(t) the algebraic closure of a field C(t) 149 Tx,p the tangent space of X at P 150

iG the Geiser involution 150 iB the Bertini involution 151 --- a birational map 152 birat

¢:X+ ---+ Y the flop of a flopping contraction ¢:X ---+ Y 158 Type(I),(II),(III),(IV)

the types of links in the Sarkisov program 159 Z,(XjS) the group of I-cycles whose support maps

to points on S 160 NI(XjS) = {Pic(X)j =s I 0;;;:: IR 162 NI(XjS) = {ZI(XjS)j =sl 0z IR 162

=s the numerical equivalence induced by the pairing Pic(X) x Z I (X j S) ---+ z 162

NE(XjS) the cone generated by effective I-cycles on X mapped to points by 7r 162

NE(XjS) the closure ofNE(XjS) in NI(XjS) 162 Sing(X) the singular locus of X 164

Xreg the nonsingular (regular) locus of X 164 Qn

v the sheaf of regular n-forms on V 165 Rldl = EBm>oH°(V, Oy{mdKv », where

R = Rill = EBm~oHo(V, Ov(mdKv» 170 Ox(l) the restriction of the tautological line bundle

OjpN (I) to X C ]P'N 171 Ie log canonical singularities 174 It log terminal singularities 174

dlt divisorially log terminal singularities 174 wklt weakly kawamata log terminal singularities 174

aCE; X, D) the discrepancy of a log pair (X, D) at E 179 beE; X, D) the log discrepancy of a log pair (X, D) at E 179

inf the infimum 179

xviii List of Notation

discrep( X, D) = inf (a(E; X, D)} 179 Centerx E the center of E on X 179

tenn terminal singularities 182 can canonical singularities 182 kIt kawamata log tenninal singularities 182 pIt purely log tenninal singulaxities 182

c:X ---+ X the canonical cover of X 183 Jc-lm ·0-p x the radical of an ideal c-lm p . Ox 185

Nonn(h) the nonn of the element h E C(X) over C(X) 186 r.(UFj ) the cycle-theoretic image of U F j by a morphism r 188

vr(a) the (discrete) valuation of a function a at a divisor r 189 vr(S) the (discrete) valuation of a section s of a line bundle

L at a divisor I' 189 gcd the greatest common divisor 190

Gal(C(X)/C(X» the Galois group of C(X) over C(X) 190 Gr the decomposition group of r, i.e.,

G r = {a E Gal(C(X)/C(X»; a r = r} 190 (C.OV)f. the stalk of a sheaf c*Ov at the generic point of E 190

{(c*Oy)!dGI" the Gr-invariant part of (c*Oy)r; 190 7r:5 ---+ 5 the minimal resolution of 5 194

f*D the pullback of a divisor D in the sense of Mumford 195 All' DII • £6. £7. £8

the types of canonical singularities in dimension 2 with their corresponding defining equations and dual graphs 197

analytically 198 "-' analytically isomorphic 203

niH." the completion of mH." 204 Cnt ]] the formal power series ring 204

embeddimp 5 the embedding dimension of 5 at p 204 C(g) the cubic part of g 208

SL(2. C) the special linear group of 2 x 2 matrices with entries in C having detenninant 1 212

CII +1 the cyclic group of order n + 1 213 DII - 2 the binary extension of the dihedral group D,,-2,

which is the group of symmetries of the regular (n - 2)-gon 213

T the binary extension of the tetrahedral group 213 0 the binary extension of the octahedral group 213 I the binary extension of the icosahedral group 213

SU(2. C) c SL(2. C) the special unitary group 213

(Note that the notation for the groups in the table of Corollary 4-6-16 on Page 212 is different from that in the table of Theorem 4-6-20 on Pages 219 and 220.)

List of Notation xix

SO(3.lR) Iz

Jl'l (S - (p}) Z/(r)

(0 E ( 2)/G

GL(n. C)

c GL(3, lR) the special orthogonal group the 2 x 2 identity matrix the (local) fundamental group of a genn S - {p} the quotient of Z by (r) the quotient of 0 E C2 by a finite group G

~ CD the transpose of (z" ... , z.J the general linear group of n x n matrices with entries in C the group of sth roots of unity the multiplicative group of C - {O}

{(experrR) O)} . n wIth o exp errq~)

o < q < nand gcd(n, q) = 1 the cyclic group of order k in ZL(2, C), the center ofGL(2, C) the cyclic group of order k in SL(2, C) the binary dihedral group of order 4k the binary tetrahedral group

o the binaxy octahedral group I the binary icosahedral group

(HI. N I ; H2. N2) = (h E GL(2. C); h = h Ih2 for some hiE HI. h2 E H2 such that hi mod NI = h2 mod N2}.

(n. q) the dual graph corresponding to the continued fraction of ~

q

(b;nl. ql;n2. Q2;··. ;n.\ .• Qs) the dual graph consisting of the center point with self-intersection number b and the branches (ni. Qi)

{ a primitive rth root of unity An. eAn• Dn. eD4 • E6 • E 7 • E8

O;.P fw !!.L ilv

analytical! y "-'

equivariant

E

the types of the log tenninal singularities according to the corresponding equivariant defining equations of the canonical cover the group of units (invertible elements) in a ring Os.P the Weierstrass fonn of a holomorphic function f the partial derivative of a function f with respect to a variable v

analytically isomorphic and equivariant

the order of an element a in a group Z/(r) = C/(Z + w - Z) the elliptic curve obtained as the quotient of C by the lattice Z + w . Z

213 213 215 215 215

216

216 225 217

219

219 219 219 219 219 219

219

220

221 222

222 224 225

226

227

233

240

xx List of Notation

7r*Oz EBr=c/7r*Oz[EiJ, the decomposition of a sheaf 7r*Oz under the cyclic group action into the eigenspaces with eigenvalues Ei 252

i,CX-D,i the extension of a sheaf CX-D,i over X - D to the whole X by setting every stalk at p E D to be zero 254

CX-D,i a locally constant sheaf of rank 1 254 Hi(X,7r*CZ )[E i ] the eigenspace of Hi(X, 7r*Cz) with eigenvalue Ei 254

C(ll, CX - D ) the tech complex of a sheaf C X - D with respect to an open covering II 255

Hi (C(ll, C X - D » the jth cohomology ofthe tech complex C(U, C XD ) 255 rakl the round up of the rational number ak, i.e.,

ak < r ak l < ak + 1 and fak 1 E Z 260 [ad the round down of the rational number ak i.e.,

ak - 1 < [ak] :::: ak and [ak] E Z 260 r Al = L r ak lAb the round up of the Q-divisor

A = LakAk 260 [A] = L[ak]Ab the round down of the Q-divisor

A = LakAk 260 HP(Y, Ky +7r*A)G

the G-invariant part of a cohomology group HI'(Y, K y + Jr* A) with G-action 261

(Jr.Oy(K y + Jr* A)}G the G-invariant part of a sheaf Jr*Oy(K y + 7r* A) 264 with G-action

O

List of Notation xxi

Type(£1), (£2), (£3), (£4), (£5) the five types of extremal contractions of DIVISO-RIAL type on nonsingular projective 3-folds 309

Type(C1), (C2 ), (D 1), (D2 ), (D3 ) the five types of extremal contractions of FIBERING type on nonsingular projective 3-folds 314

V2 the double cover of JID2 whose branch locus is a

Grass(2,5)

p(X/y)

im{F ---* m d(X)

char(k) Fm:C ---* C

Def(j, B)

dimLIIDef(j, B)

u: UnivJi ---* H

H rat

birational

smooth quartic surface the grassmanian variety parametrizing 2-dimensional subspaces in the 5-dimensional ambient vector space = dimN 1 (X / Y) the relative Picard number of X over Y the group of Weil divisors on n-dimensional variety X considered as (n - I)-cycles the image of a sheaf homomorphism F ---* 9 the "difficulty" of X the characteristic of a field k the Frobenius morphism of degree q = pm the space parametrizing the deformations of a morphism f:C ---* X fixing a subscheme B the dimension of the deformation space at the point [fl corresponding to the original morphism f the universal family over a subspace H of the Hilbert scheme = {the closure of {p E H; u-1(p) is an irreducible rational curve}

315

315

318

320 321 322 325 325

325

325

328

328

- - ---> a birational map 334 dominant

---* a dominant morphism 334 :D the category of logarithmic pairs consisting of normal

projective varieties and boundary Q-divisors with only Q-factorial and log terminal singularities 341

~ the category of logarithmic pairs consisting of non-singular projective varieties and boundary Q-divisors with only normal crossings 341

(X, D)can = Proj EBm~o HO(X, m(Kx + D» the log canonical model of a log pair (X, D) 345

CQ = {XY - ZW = O} c {(X: Y : Z : Y : T)} = jp>4, the cone over a smooth quadric 350

multcv , D the multiplicity of a divisor D along (the generic point of) a curve Cv,i 355

b(D,

xxii List of Notation

bmin = min{b(D, Q>i)} I

D bmin ~ D = L.bmin=l-dk k Div(X I) the group of Cartier divisors on X I

[D] the numerical class of a divisor D in N I (X) Mov = the convex cone generated by the classes

represented by the movable divisors over Xcan Mov = the closure of Mov

MovQ = Mov n Nb where Nb = {Pic (X)/ =X.aJ ® Q Amp(X / Xcan) the nef cone over Xcan i.e., the closure of the cone

generated by ample divisors over X can

A I the affine Weyl group of type A I X'I = X X y Spec k( y), where 1] = Spec k( Y) is the generic

point of Y Z( Vi. Xi) the center of a discrete valuation Vi on Xi

e(z(v;. Xi); Xi. llx,) the canonical threshold of the pair (Xi. llx;) at the cen-ter Z(Vi, Xi), i.e., = maxIe E Q~o; aCE; X. ellx,) ~ 0 for all the discrete valuations with the center on Xi being Z( Vi, X;)}

::D(W.Bw ) the category of all log pairs obtained from a log surface (W. Bw) via K + B-MMP

T(k, l)

B(k, l)

C* N ~Zn

M

the toric projective surface defined by the complete fan (in NR = N ® JR. = 'Z} ® JR.) whose I-dimensional cones are defined by (k, I), (0, I) and (-I, - 1) where k > I ~ 0 and gcd(k. I) = I unless (k, I) = (I. 0) the (closed) divisor corresponding to the I-dimensional cone spanned by (k, I) the Hirzebruch surface having the unique section ai with the minimum self-intersection a/ = -I = ]p>1 - to, Do} a lattice of rank n representing the I-parameter subgroups Hom( G m , TN) Homz(N, Z), the dual lattice of N representing the characters Hom( TN, G m) a fan in NB, the toric variety associated to a fan t. a (strictly convex) rational polyhedral cone

= N ®z JR. (VI,.'" vs ) = {rivi + ... + rsvs;ri ~ O} the cone generated by

355

356 358 359

361 361 361

361 366

386 397

397

400

402

402

407 408

413

413 413 413 413 413

vectors VI, ... , Vs 413 Ua = SpecC[a V n M] the affine toric variety associated

to a cone a 414 a V ={uEM'R,.=M®JR.;u(v)~O VVEa},

the dual cone of a c N'R,. 414

TN O(y) V(y)

Av N(a)

M(a) Stab(xw)

List of Notation xxiii

= k* = Spec k[t, t- I ], the multiplicative group of the base field k ~ (Gmt the n-dimensional algebraic torus the orbit corresponding to a cone y the closure of the orbit corresponding to a cone y the rational equivalence of cycles the I-parameter subgroup corresponding to v E N = N / Nu • where Nu = N n (the subvector space spanned by a) = a-.l n M = Hom (N(a), Z). the dual of N(a) = {t' E T'; t' . Xw = xw}, the stabilizer group of a point Xw in the torus T' the group of I-cycles modulo rational equivalence on X(6) the group of (n - I )-cycles modulo rational (linear) equivalence on an n-dimensional variety X(6)

414 414 414 414 416 416

416 416

416

418 418

(vf, ...• v;, ... , V~_I' v;) the basis of MJR dual to a basis (VI, ..• , Vj, ... , Vn-I, vn ) of NJR such that V;(Vj) = Ojj 421

xm the rational function corresponding to m E M 421 CPR the contraction morphism of an extremal ray R 422

JP'(do : ... : dn - fl ) the weighted projective space dimension n - f3 with weight (do . ... , dn - fl ) 433

Ix} = x - [x 1 the fractional part of a number x 437 (~), the simplex that is the convex hull of the origin and

the primitive vectors generating a cone r 441 Vol the volume with respect to the usual Euclidean

metric in NJR 441 m, E M 0 Q such that m,(vj) = 1 VVj E N primitive

with (Vj) E 6 and (Vj) C r 442 ordv D the order (coefficient) of a prime divisor V in a

Weil divisor D 444 o the d-dimensional simplex whose vertices are the

IkA - LI lc(X(6), D)

e(X(6), D)

vectors Vo, VI, ...• Vd the distance between points k A and W the log canonical threshold of X(6) with respect to D = maxie E Q>o;a(E; X(6), eD) ::: -I VE exceptional divisors} the canonical threshold of X (6) with respect to D = maxie E Q>o; aCE; X(6), eD) ::: 0 V E exceptional divisors}

450 451

452

454