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Publisher Summary 1
Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, J谩nos Koll谩r 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. Koll谩r 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.
目录
Introduction p. 1
Resolution for Curves p. 5
Newton's method of rotating rulers p. 5
The Riemann surface of an algebraic function p. 9
The Albanese method using projections p. 12
Normalization using commutative algebra p. 20
Infinitely near singularities p. 26
Embedded resolution, I: Global methods p. 32
Birational transforms of plane curves p. 35
Embedded resolution, II: Local methods p. 44
Principalization of ideal sheaves p. 48
Embedded resolution, III: Maximal contact p. 51
Hensel's lemma and the Weierstrass preparation theorem p. 52
Extensions of K((t)) and algebroid curves p. 58
Blowing up 1-dimensional rings p. 61
Resolution for Surfaces p. 67
Examples of resolutions p. 68
The minimal resolution p. 73
The Jungian method p. 80
Cyclic quotient singularities p. 83
The Albanese method using projections p. 89
Resolving double points, char p. 6=
Embedded resolution using Weierstrass' theorem p. 101
Review of multiplicities p. 110
Strong Resolution in Characteristic Zero p. 117
What is a good resolution algorithm? p. 119
Examples of resolutions p. 126
Statement of the main theorems p. 134
Plan of the proof p. 151
Birational transforms and marked ideals p. 159
The inductive setup of the proof p. 162
Birational transform of derivatives p. 167
Maximal contact and going down p. 170
Restriction of derivatives and going up p. 172
Uniqueness of maximal contact p. 178
Tuning of ideals p. 183
Order reduction for ideals p. 186
Order reduction for marked ideals p. 192
Bibliography p. 197
Index p. 203
Resolution for Curves p. 5
Newton's method of rotating rulers p. 5
The Riemann surface of an algebraic function p. 9
The Albanese method using projections p. 12
Normalization using commutative algebra p. 20
Infinitely near singularities p. 26
Embedded resolution, I: Global methods p. 32
Birational transforms of plane curves p. 35
Embedded resolution, II: Local methods p. 44
Principalization of ideal sheaves p. 48
Embedded resolution, III: Maximal contact p. 51
Hensel's lemma and the Weierstrass preparation theorem p. 52
Extensions of K((t)) and algebroid curves p. 58
Blowing up 1-dimensional rings p. 61
Resolution for Surfaces p. 67
Examples of resolutions p. 68
The minimal resolution p. 73
The Jungian method p. 80
Cyclic quotient singularities p. 83
The Albanese method using projections p. 89
Resolving double points, char p. 6=
Embedded resolution using Weierstrass' theorem p. 101
Review of multiplicities p. 110
Strong Resolution in Characteristic Zero p. 117
What is a good resolution algorithm? p. 119
Examples of resolutions p. 126
Statement of the main theorems p. 134
Plan of the proof p. 151
Birational transforms and marked ideals p. 159
The inductive setup of the proof p. 162
Birational transform of derivatives p. 167
Maximal contact and going down p. 170
Restriction of derivatives and going up p. 172
Uniqueness of maximal contact p. 178
Tuning of ideals p. 183
Order reduction for ideals p. 186
Order reduction for marked ideals p. 192
Bibliography p. 197
Index p. 203
Lectures on resolution of singularities /
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