Flips for 3-folds and 4-folds

by Corti, Alessio

• ISBN13: 9780198570615
• ISBN10: 0198570619
• Format: Hardcover
• Copyright: 2007-08-23
• Publisher: Oxford University Press

Summary

This edited collection of chapters, authored by leading experts, provides a complete and essentially self-contained construction of 3-fold and 4-fold klt flips. A large part of the text is a digest of Shokurov's work in the field and a concise, complete and pedagogical proof of the existence of 3-fold flips is presented. The text includes a ten page glossary and is accessible to students and researchers in algebraic geometry.

Table of Contents

List of Contributors	p. ix
Introduction	p. 1
Minimal models of surfaces	p. 1
Higher dimensions and flips	p. 1
The work of Shokurov	p. 2
Minimal models of 3-folds and 4-folds	p. 3
The aim of this book	p. 3
Pl flips	p. 4
b-divisors	p. 5
Restriction and mobile b-divisors	p. 6
Pbd-algebras	p. 6
Restricted systems and 3-fold pl flips	p. 7
Shokurov's finite generation conjecture	p. 8
What is log terminal?	p. 9
Special termination and reduction to pl flips	p. 10
The work of Hacon and McKernan: adjoint algebras	p. 10
Saturated mobile b-divisors on weak del Pezzo klt surfaces	p. 14
The CCS Conjecture	p. 14
Kodaira's canonical bundle formula and adjunction	p. 14
Finite generation on non-klt surfaces	p. 15
The glossary	p. 15
The book as a whole	p. 15
Termination?	p. 16
The work of Siu	p. 16
Pre-requisites	p. 16
3-fold flips after Shokurov	p. 18
Introduction	p. 18
Statement and brief history of the problem	p. 18
Summary of the chapter	p. 19
Other surveys	p. 20
Background	p. 20
Summary	p. 20
Discrepancy	p. 20
Klt and plt	p. 22
Inversion of adjunction	p. 22
The flip conjecture	p. 23
Reduction of klt flips to pl flips	p. 23
Plan of the proof	p. 24
Log resolution	p. 25
The language of function algebras and b-divisors	p. 26
Function algebras	p. 26
b-divisors	p. 27
Saturated divisors and b-divisors	p. 30
Mobile b-divisors	p. 32
Restriction and saturation	p. 34
Pbd-algebras: terminology and first properties	p. 36
The limiting criterion	p. 37
Function algebras and pbd-algebras	p. 37
The finite generation conjecture	p. 38
A Shokurov algebra on a curve is finitely generated	p. 39
Finite generation on surfaces and existence of 3-fold flips	p. 40
The finite generation conjecture implies existence of pl flips	p. 40
Linear systems on surfaces	p. 42
The finite generation conjecture on surfaces	p. 45
What is log terminal?	p. 49
What is log terminal?	p. 49
Preliminaries on Q-divisors	p. 50
Singularities of pairs	p. 52
Divisorially log terminal	p. 52
Resolution lemma	p. 53
Whitney umbrella	p. 54
What is a log resolution?	p. 56
Examples	p. 58
Adjunction for dlt pairs	p. 59
Miscellaneous comments	p. 60
Special termination and reduction to pl flips	p. 63
Introduction	p. 63
Special termination	p. 64
Reduction theorem	p. 68
The non-Q-factorial minimal model program	p. 72
Extension theorems and the existence of flips	p. 76
Introduction	p. 76
The conjectures of the MMP	p. 76
Previous results	p. 78
Sketch of the proof	p. 78
Notation and conventions	p. 82
The real minimal model program	p. 83
Finite generation	p. 87
Generalities on finite generation	p. 87
Adjoint algebras	p. 89
Basic properties of the mobile part	p. 91
Multiplier ideal sheaves and extension results	p. 95
Definition and first properties of multiplier ideal sheaves	p. 95
Extending sections	p. 100
The restricted algebra is an adjoint algebra	p. 107
Saturated mobile b-divisors on weak del Pezzo klt Surfaces	p. 111
Introduction	p. 111
Example	p. 112
Preliminaries	p. 113
Mobile b-divisors of Iitaka dimension one	p. 115
Confined divisors	p. 121
Introduction	p. 121
Diophantine approximation and descent	p. 122
Confined b-divisors	p. 124
The CCS conjecture	p. 126
The surface case	p. 128
A strategy for the general case	p. 129
Kodaira's Canonical bundle formula and adjunction	p. 134
Introduction	p. 134
Fujita's canonical bundle formula	p. 136
The general canonical bundle formula	p. 138
Fibre spaces of log Kodaira dimension 0	p. 142
Kawamata's canonical bundle formula	p. 148
Adjunction	p. 150
Tie breaking	p. 152
Log canonical purity	p. 154
Ambro's seminormality theorem	p. 156
Covering tricks and semipositivity of f*¿¿/Y	p. 158
Non-klt techniques	p. 163
Introduction	p. 163
Preliminary	p. 163
Codimension one adjunction	p. 163
The discriminant of a log pair	p. 164
Non-klt finite generation	p. 166
Glossary	p. 171
Bibliography	p. 181
Index	p. 187
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