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.

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 6= 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

János Kollár is a professor of Mathematics at Princeton University.