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Lectures on Resolution of Singularities

János Kollár
Princeton University Press
Publication Date: 
Number of Pages: 
Annals of Mathematics Studies 166
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Darren Glass
, on

Algebraic geometers like to study varieties — those subsets of projective space that can be defined by polynomial equations. And in many applications within algebraic geometry smooth points — that is, points where the implicit function theorem works or points where not all the partial derivatives vanish — behave much more nicely than singular points — those points which are not smooth, such as nodes and cusps. Because of this, many people have developed techniques to deal with singularities that can be broadly classified as the 'resolution of singularities', which roughly speaking takes an arbitrary variety X which may have singular points and defines a variety X' which does not have any singular points so that there is a nice surjection mapping X' to X. There are many ways of approaching this problem, each of which comes with various pros and cons, and in János Kollár's new book Lectures on Resolution of Singularities he gives a comprehensive and very readable overview of the topic.

The book is based on lectures Kollár gave at Princeton and at the University of Utah, and is divided into three chapters. The first chapter deals with the resolution of singularities on curves, and discusses thirteen distinct approaches to the problem (several of which have sub-approaches). These range from Newton's method of rotating rulers and Hensel's Lemma to the Albanese method and the so-called 'blowing up' of singularities. Some of these approaches involve lots of background and machinery from algebraic geometry, topology, and other branches of mathematics, but others are quite elementary. The second chapter takes several of these methods and generalizes to the case of surfaces, where one often needs to use more care (and at times more machinery) in order to make the program work. The third chapter deals with the general case, where one can actually make quite a bit of headway using (relatively) elementary methods as long as you are willing to limit yourself to working in characteristic zero.

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

Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Number Theory, and Cryptography. He can be reached at [email protected].

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