Lectures on Resolution of Singularities

by Kollar, Janos

• ISBN13: 9780691129235
• ISBN10: 0691129231
• Format: Paperback
• Copyright: 2007-02-05
• Publisher: Princeton Univ Pr

Summary

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.

Table of 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 sheaves	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