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Lectures on Resolution of Singularities (AM-166) : Annals of Mathematics Studies (Paperback) - Janos Kollar

Lectures on Resolution of Singularities (AM-166)

Annals of Mathematics Studies (Paperback)

Paperback Published: 25th February 2007
ISBN: 9780691129235
Number Of Pages: 208

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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.

Industry Reviews

"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

Introductionp. 1
Resolution for Curvesp. 5
Newton's method of rotating rulersp. 5
The Riemann surface of an algebraic functionp. 9
The Albanese method using projectionsp. 12
Normalization using commutative algebrap. 20
Infinitely near singularitiesp. 26
Embedded resolution, I: Global methodsp. 32
Birational transforms of plane curvesp. 35
Embedded resolution, II: Local methodsp. 44
Principalization of ideal sheavesp. 48
Embedded resolution, III; Maximal contactp. 51
Hensel's lemma and the Weierstrass preparation theoremp. 52
Extensions of K ((t)) and algebroid curvesp. 58
Blowing up 1-dimensional ringsp. 61
Resolution for Surfacesp. 67
Examples of resolutionsp. 68
The minimal resolutionp. 73
The Jungian methodp. 80
Cyclic quotient singularitiesp. 83
The Albanese method using projectionsp. 89
Resolving double points, char [Not Equal] 2p. 97
Embedded resolution using Weierstrass' theoremp. 101
Review of multiplicitiesp. 110
Strong Resolution in Characteristic Zerop. 117
What is a good resolution algorithm?p. 119
Examples of resolutionsp. 126
Statement of the main theoremsp. 134
Plan of the proofp. 151
Birational transforms and marked idealsp. 159
The inductive setup of the proofp. 162
Birational transform of derivativesp. 167
Maximal contact and going downp. 170
Restriction of derivatives and going upp. 172
Uniqueness of maximal contactp. 178
Tuning of idealsp. 183
Order reduction for idealsp. 186
Order reduction for marked idealsp. 192
Bibliographyp. 197
Indexp. 203
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691129235
ISBN-10: 0691129231
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 208
Published: 25th February 2007
Country of Publication: US
Dimensions (cm): 22.07 x 15.04  x 1.45
Weight (kg): 0.31

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