Pub. Date:
Princeton University Press
Lectures on Resolution of Singularities (AM-166)

Lectures on Resolution of Singularities (AM-166)

by János Kollár
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Product Details

ISBN-13: 9780691129235
Publisher: Princeton University Press
Publication date: 02/25/2007
Series: Annals of Mathematics Studies , #179
Edition description: New Edition
Pages: 208
Product dimensions: 6.00(w) x 9.25(h) x (d)

About the Author

János Kollár is a professor of Mathematics at Princeton University.

Table of Contents

Introduction     1
Resolution for Curves     5
Newton's method of rotating rulers     5
The Riemann surface of an algebraic function     9
The Albanese method using projections     12
Normalization using commutative algebra     20
Infinitely near singularities     26
Embedded resolution, I: Global methods     32
Birational transforms of plane curves     35
Embedded resolution, II: Local methods     44
Principalization of ideal sheaves     48
Embedded resolution, III; Maximal contact     51
Hensel's lemma and the Weierstrass preparation theorem     52
Extensions of K ((t)) and algebroid curves     58
Blowing up 1-dimensional rings     61
Resolution for Surfaces     67
Examples of resolutions     68
The minimal resolution     73
The Jungian method     80
Cyclic quotient singularities     83
The Albanese method using projections     89
Resolving double points, char [Not Equal] 2     97
Embedded resolution using Weierstrass' theorem     101
Review of multiplicities     110
Strong Resolution in Characteristic Zero     117
What is a good resolution algorithm?     119
Examples of resolutions     126
Statement of the main theorems     134
Plan of the proof     151
Birational transforms and marked ideals     159
The inductive setup of the proof     162
Birational transform of derivatives     167
Maximal contact and going down     170
Restriction of derivatives and going up     172
Uniqueness of maximal contact     178
Tuning of ideals     183
Order reduction for ideals     186
Order reduction for marked ideals     192
Bibliography     197
Index     203

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