Lectures on Resolution of Singularities (AM-166)

Lectures on Resolution of Singularities (AM-166)

by Janos Kollar
     
 

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ISBN-10: 0691129223

ISBN-13: 9780691129228

Pub. Date: 02/05/2007

Publisher: Princeton University Press

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

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:
9780691129228
Publisher:
Princeton University Press
Publication date:
02/05/2007
Series:
Annals of Mathematics Studies Series
Pages:
198
Product dimensions:
6.30(w) x 9.30(h) x 0.80(d)

Table of Contents

Contents

Introduction....................1
Chapter 1. Resolution for Curves....................5
1.1. Newton's method of rotating rulers....................5
1.2. The Riemann surface of an algebraic function....................9
1.3. The Albanese method using projections....................12
1.4. Normalization using commutative algebra....................20
1.5. Infinitely near singularities....................26
1.6. Embedded resolution, I: Global methods....................32
1.7. Birational transforms of plane curves....................35
1.8. Embedded resolution, II: Local methods....................44
1.9. Principalization of ideal sheave....................48
1.10. Embedded resolution, III: Maximal contact....................51
1.11. Hensel's lemma and the Weierstrass preparation theorem....................52
1.12. Extensions of K((t)) and algebroid curves....................58
1.13. Blowing up 1-dimensional rings....................61
Chapter 2. Resolution for Surfaces....................67
2.1. Examples of resolutions....................68
2.2. The minimal resolution....................73
2.3. The Jungian method....................80
2.4. Cyclic quotient singularities....................83
2.5. The Albanese method using projections....................89
2.6. Resolving double points, char [not equal to] 2....................97
2.7. Embedded resolution using Weierstrass' theorem....................101
2.8. Review of multiplicities....................110
Chapter 3. Strong Resolution in Characteristic Zero....................117
3.1. What is a good resolution algorithm?....................119
3.2. Examples of resolutions....................126
3.3. Statement of the main theorems....................134
3.4. Plan of the proof....................151
3.5. Birational transforms and marked ideals....................159
3.6. The inductive setup of the proof....................162
3.7. Birational transform of derivatives....................167
3.8. Maximal contact and going down....................170
3.9. Restriction of derivatives and going up....................172
3.10. Uniqueness of maximal contact....................178
3.11. Tuning of ideals....................183
3.12. Order reduction for ideals....................186
3.13. Order reduction for marked ideals....................192
Bibliography....................197
Index....................203

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