Lectures on Resolution of Singularities

by János Kollár

Overview

Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

Product Details

ISBN-13:	9780691129235
Publisher:	Princeton University Press
Publication date:	02/25/2007
Series:	Annals of Mathematics Studies , #166
Edition description:	New Edition
Pages:	208
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

János Kollár is a professor of Mathematics at Princeton University.

Table of Contents

Introduction     1
Resolution for Curves     5
Newton's method of rotating rulers     5
The Riemann surface of an algebraic function     9
The Albanese method using projections     12
Normalization using commutative algebra     20
Infinitely near singularities     26
Embedded resolution, I: Global methods     32
Birational transforms of plane curves     35
Embedded resolution, II: Local methods     44
Principalization of ideal sheaves     48
Embedded resolution, III; Maximal contact     51
Hensel's lemma and the Weierstrass preparation theorem     52
Extensions of K ((t)) and algebroid curves     58
Blowing up 1-dimensional rings     61
Resolution for Surfaces     67
Examples of resolutions     68
The minimal resolution     73
The Jungian method     80
Cyclic quotient singularities     83
The Albanese method using projections     89
Resolving double points, char [Not Equal] 2     97
Embedded resolution using Weierstrass' theorem     101
Review of multiplicities     110
Strong Resolution in Characteristic Zero     117
What is a good resolution algorithm?     119
Examples of resolutions     126
Statement of the main theorems     134
Plan of the proof     151
Birational transforms and marked ideals     159
The inductive setup of the proof     162
Birational transform of derivatives     167
Maximal contact and going down     170
Restriction of derivatives and going up     172
Uniqueness of maximal contact     178
Tuning of ideals     183
Order reduction for ideals     186
Order reduction for marked ideals     192
Bibliography     197
Index     203

Editorial Reviews

"Throughout his lectures, Kollar uses plenty of motivations and examples, and the text is very readable. Any graduate student or mathematicians who wishes to learn about the subject would be well-served to use this book as a starting point."—-Darren Glass, MAA Review

"People are already using this book. I am using this book now. I expect it will be used well into the future."—-Dan Abramovich, Mathematical Reviews

"The book will be an invaluable tool not only for graduate student, but also for algebraic geometers. Mathematicians working in different fields will also enjoy the clarity of the exposition and the wealth of ideas included. This will become, I'm sure, as it happened to most books in this series, one of the classics of modern mathematics."—-Paul Blaga, Mathematica

From the Publisher

The book will be an invaluable tool not only for graduate student, but also for algebraic geometers. Mathematicians working in different fields will also enjoy the clarity of the exposition and the wealth of ideas included. This will become, I'm sure, as it happened to most books in this series, one of the classics of modern mathematics.

Mathematica - Paul Blaga

People are already using this book. I am using this book now. I expect it will be used well into the future.

Mathematical Reviews - Dan Abramovich

Throughout his lectures, Kollar uses plenty of motivations and examples, and the text is very readable. Any graduate student or mathematicians who wishes to learn about the subject would be well-served to use this book as a starting point.

MAA Review - Darren Glass

The book will be an invaluable tool not only for graduate student, but also for algebraic geometers. Mathematicians working in different fields will also enjoy the clarity of the exposition and the wealth of ideas included. This will become, I'm sure, as it happened to most books in this series, one of the classics of modern mathematics.
— Paul Blaga

Mathematica

People are already using this book. I am using this book now. I expect it will be used well into the future.
— Dan Abramovich

Mathematical Reviews

Throughout his lectures, Kollar uses plenty of motivations and examples, and the text is very readable. Any graduate student or mathematicians who wishes to learn about the subject would be well-served to use this book as a starting point.
— Darren Glass

MAA Review