Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, Janos Kollar 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. Kollar 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.
Janos Kollar is a professor of Mathematics at Princeton University.
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
"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
LECTURES ON RESOLUTION OF SING
Princeton University Press
Annals of Mathematics Studies (Paperback)
Princeton University Press
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