Lectures on Resolution of Singularities (AM-166) (Annals of Mathematics Studies) (Annals of Mathematics Studies, 166, Band 166) - Softcover

Kollar, Janos

Softcover

ISBN 10: 0691129231 ISBN 13: 9780691129235

Verlag: Princeton University Press, 2007

Inhaltsangabe

Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár 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. Kollár 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.

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János Kollár

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Lectures on Resolution of Singularities

By Jnos Kollr

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-12923-5

Contents

Introduction.....................................................................1Chapter 1. Resolution for Curves.................................................51.1. Newton's method of rotating rulers..........................................51.2. The Riemann surface of an algebraic function................................91.3. The Albanese method using projections.......................................121.4. Normalization using commutative algebra.....................................201.5. Infinitely near singularities...............................................261.6. Embedded resolution, I: Global methods......................................321.7. Birational transforms of plane curves.......................................351.8. Embedded resolution, II: Local methods......................................441.9. Principalization of ideal sheave............................................481.10. Embedded resolution, III: Maximal contact..................................511.11. Hensel's lemma and the Weierstrass preparation theorem.....................521.12. Extensions of K((t)) and algebroid curves..................................581.13. Blowing up 1-dimensional rings.............................................61Chapter 2. Resolution for Surfaces...............................................672.1. Examples of resolutions.....................................................682.2. The minimal resolution......................................................732.3. The Jungian method..........................................................802.4. Cyclic quotient singularities...............................................832.5. The Albanese method using projections.......................................892.6. Resolving double points, char [not equal to] 2..............................972.7. Embedded resolution using Weierstrass' theorem..............................1012.8. Review of multiplicities....................................................110Chapter 3. Strong Resolution in Characteristic Zero..............................1173.1. What is a good resolution algorithm?........................................1193.2. Examples of resolutions.....................................................1263.3. Statement of the main theorems..............................................1343.4. Plan of the proof...........................................................1513.5. Birational transforms and marked ideals.....................................1593.6. The inductive setup of the proof............................................1623.7. Birational transform of derivatives.........................................1673.8. Maximal contact and going down..............................................1703.9. Restriction of derivatives and going up.....................................1723.10. Uniqueness of maximal contact..............................................1783.11. Tuning of ideals...........................................................1833.12. Order reduction for ideals.................................................1863.13. Order reduction for marked ideals..........................................192Bibliography.....................................................................197Index............................................................................203

Chapter One

Resolution for Curves

Resolution of curve singularities is one of the oldest and prettiest topics of algebraic geometry. In all likelihood, it is also completely explored.

In this chapter I have tried to collect all the different ways of resolving singularities of curves. Each of the thirteen sections contains a method, and some of them contain more than one. These come in different forms: solving algebraic equations by power series, normalizing complex manifolds, projecting space curves, blowing up curves contained in smooth surfaces, birationally transforming plane curves, describing field extensions of Laurent series fields and blowing up or normalizing 1-dimensional rings.

By the end of the chapter we see that the methods are all interrelated, and there is only one method to resolve curve singularities. I found, however, that these approaches all present a different viewpoint or technical twist that is worth exploring.

1.1. Newton's method of rotating rulers

Let F(x, y) be a complex polynomial in two variables. We are interested in finding solutions of F = 0 in the form y = [phi](x), where [phi] is some type of function that we are right now unsure about.

Following the classical path of solving algebraic equations, one might start with the case where [phi](x) is a composition of polynomials, rational functions and various mth roots of these. As in the classical case, this will not work if the degree of F is 5 or more in y. One can also try to look for power series solutions, but simple examples show that we have to work with power series with fractional exponents. The equation [y.sup.m] = x + [x.sup.2] has no power series solutions for m [greater than or equal to] 2, but it has fractional power series solutions for any [[epsilon].sup.m] = 1 given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a more interesting example, [y.sup.m] - [y.sup.n] + x = 0 for m > n also has a fractional power series solution

y = [[summation].sub.i]greater than or equal to]1] [a.sub.i][x.sup.i/n],

where [a.sub.1] = 1 and the other [a.sub.i] are defined recursively by

n [a.sub.s] = coefficient of [x.sup.(s+n-1)/n] in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

After many more examples, we are led to look for solutions of the form

y = [[infinity].summation over (i=0)] [c.sub.i][x.sup.i/M],

where M is a natural number whose dependence on deg F we leave open for now. These series, though introduced by Newton, are called Puiseux series. We encounter them later several times.

Theorem 1.1 (Newton, 1676). Let F(x, y) be a complex polynomial or power series in two variables. Assume that F(0, 0) = 0 and that [y.sup.n] appears in F(x, y) with a nonzero coefficient for some n. Then F(x, y) = 0 has a Puiseux series solution of the form

y = [[infinity].summation over (i=1)] [c.sub.i][x.sup.i/N]

for some integer N.

Remark 1.2. (1) The original proof is in a letter of Newton to Oldenburg dated October 24, 1676. Two accessible sources are [New60, pp.126-127] and [BK81, pp.372-375].

(2) Our construction gives only a formal Puiseux series; that is, we do not prove that it converges for [absolute value of x] sufficiently small. Nonetheless, if F is a polynomial or a power series that converges in some neighborhood of the origin, then any Puiseux series solution converges in some (possibly smaller) neighborhood of the origin. This is easiest to establish using the method of Riemann, to be discussed in Section 1.2.

(3) By looking at the proof we see that we get n different solutions (when counted with multiplicity).

The proof of Newton starts with a graphical representation of the "lowest order" monomials occurring in F. This is now called the Newton polygon.

Definition 1.3 (Newton polygon). Let F =...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2007
• Sprache: Englisch
• ISBN 10: 0691129231
• ISBN 13: 9780691129235
• Einband: Taschenbuch
• Anzahl der Seiten: 216
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: gpsr@libri.de; gpsr@libri.de; ; Friedensallee 273; Hamburg; 22763; Deutschland