Resolution of Singularities: A research textbook in tribute to Oscar Zariski Based on the courses given at the Working Week in Obergurgl, Austria, September 7–14, 1997

610 Pages·2000·44.835 MB·English

by  Joseph Lipman (auth.), Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quirós (eds.)

Progress in Mathematics
Volume 181
Series Editors
H. Bass
J. Oesterle
A. Weinstein
Resolution of
Singularities
A research textbook in tribute to
Oscar Zariski
Based on the courses given at the Working Week in
Obergurgl, Austria, September 7-14, 1997
H. Hauser
J. Lipman
F. Oort
A. Quirös
Editors
Springer Basel AG
Editors:
Herwig Hauser Joseph Lipman
Institut für Mathematik Department of Mathematics
Universität Innsbruck Purdue University
A-6020 Innsbruck West Lafayette, IN 47907
USA
Frans Oort Adolfo Quirös
Department of Mathematics Departamento de Matemäticas
Universiteit Utrecht Universidad Autönoma de Madrid
3508 TA Utrecht 28049 Madrid
The Netherlands Spain
1991 Mathematics Subject Classification 14B05, 32Sxx, 58C27
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Resolution of singularities: a research textbook in tribute to Oscar Zariski; based on the
courses given at the working week in Obergurgel, Austria, September 7- 14, 1997 /
H. Hauser ... - Boston; Basel; Berlin: Birkhäuser, 2000
(Progress in mathematics; Vol. 181)
ISBN 978-3-0348-9550-7 ISBN 978-3-0348-8399-3 (eBook)
DOI 10.1007/978-3-0348-8399-3
ISBN 978-3-0348-9550-7
This work is subject to copyright. All rights are reserved, whether the whole or part of the
material is concerned, specifically the rights of translation, reprinting, re-use of
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© 2000 Springer Basel AG
Originally published by Birkhäuser Verlag in 2000
Softcover reprint of the hardcover 1 st edition 2000
Printed on acid-free paper produced of chlorine-free pulp. TCF oo
ISBN 978-3-0348-9550-7
987654321
Table of Contents
Preface ................................................ Xl
Program of the Working Week on Resolution of Singularities ......... xiii
Participants of the Working Week Xv
J. Lipman
Oscar Zariski 1899-1986 1
H. Hauser
Resolution of Singularities 1860-1999 5
Main achievements .................................... 5
Some research problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Dictionary .......................................... 15
Surveys ............................................ 25
Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
Selection of references before 1930 ......................... 26
Selection of references after 1930 .......................... 27
Part 1: Classes of the Working Week
D. Abramovich, F. Oort
Alterations and resolution of singularities ....................... 39
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Part I The alteration theorem ............................... 45
1 Some preliminaries and generalities on varieties . . . . . . . . . . . . . . . . 45
2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Some tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Proof of de Jong's main theorem .......................... 58
5 Modifications of the proof for Theorems 2.4 and 2.7 . . . . . . . . . . . .. 67
6 Toroidal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
7 Weak resolution of singularities I .......................... 74
8 Weak resolution of singularities II . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9 Intersection multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Part II Moduli of curves ................................... 80
10 Introduction to moduli of curves .......................... 80
11 Stable reduction and completeness of moduli spaces . . . . . . . . . . . . . 86
12 Construction of moduli spaces ............................ 93
VI Table of Contents
13 Existence of tautological families .......................... 96
14 Moduli, automorphisms, and families ....................... 100
References .......................................... 104
J.-M. Aroca
Reduction of singularities for differential equations ................ 109
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 109
2 Singular foliations (co dimension one) ....................... 111
3 Reduction of singularities in dimension two . . . . . . . . . . . . . . . . . .. 115
4 Existence of integral curves .............................. 123
References .......................................... 126
J.-M. Aroca
Puiseux solutions of singular differential equations 129
1 Introduction ........................... . 129
2 The Newton polygon of a differential equation ................ . 133
3 Solutions for first order, first degree equations ................ . 139
References ......................................... . 144
S. Encinas, O. Villamayor
A course on constructive desingularization and equivariance ......... . 147
Introduction ........................................ . 147
1 First definitions ..................................... . 150
2 Pairs ............................................. . 153
3 Constructive desingularization .. . . . . . . . . . . . . . . . . . . . . . . . .. . 156
4 Basic objects ....................................... . 157
5 Monomial case .................................... .. . 165
6 Compatibility of the function t with induction ................ . 170
7 Equivariant desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 179
8 Change of base field and generalization to
the non-hypersurface case ............... . . . . . . . . . . . . . . . . 193
9 Proofs ............................................ . 200
10 Exercises: Order of ideals and upper-semi-continuity ........... . 208
11 Exercises: Blow-ups ................................... . 210
12 Exercises: Desingularization ............................. . 212
13 Exercises: On basic objects ............................. . 213
14 Exercises: A do-it-yourself help guide of theorem 7.10 .......... . 219
15 Exercises: Non-embedded constructive desingularization.
Compatibility with group actions and formal isomorphisms ...... . 221
References ......................................... . 225
Table of Contents vii
Part 2: Contributions
G. Bodnar, J. Schicho
A computer program for the resolution of singularities .............. 231
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 231
2 How to blow up a circle ............................... " 232
3 How to compute the centers of the blowups. . . . . . . . . . . . . . . . . .. 236
References .......................................... 238
V. Cossart
Uniformisation et desingularisation des surfaces d'apres Zariski ....... 239
Introduction ....................................... " 239
1 Uniformisation et recollement ........................... " 240
2 Desingularisation par eclatements normalises . . . . . . . . . . . . . . . . .. 251
References .......................................... 257
D. Cox
Toric varieties and toric resolution ............................ 259
1 Introduction ....................................... " 259
2 Cones, fans, and toric varieties .......................... " 261
3 Divisors and support functions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 266
4 Other constructions of toric varieties ....................... 270
5 Toric blow-ups and resolution of singularities ................. 276
6 Conclusion ........................................ " 283
7 Acknowledgements .................................... 283
References .......................................... 283
B. van Geemen, F. Oort
A compactification of a fine moduli space of curves ................ 285
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 285
1 Notation ........................................... 286
2 Construction of a tautological family ....................... 288
3 Local structure of the moduli space ........................ 294
References .......................................... 297
T. Geisser
Applications of de Jong's theorem on alterations .................. 299
1 Introduction ....................................... " 299
2 de Jong's theorem and Serre's conjecture .................... 300
3 Grothendieck topologies for which alterations are coverings ....... 301
4 Applications using trace maps ............................ 307
References .......................................... 313
Vlll Table of Contents
R. Goldin, B. Teissier
Resolving singularities of plane analytic branches
with one toric morphism ................................... 315
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315
2 Puiseux expansion and the semigroup of a curve ............... 317
3 Deforming curves ..................................... 319
4 Resolution using toric morphisms . . . . . . . . . . . . . . . . . . . . . . . . .. 321
5 Existence of a toric resolution ............................ 325
6 Simultaneous resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 331
7 The transforms of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . .. 334
8 Appendix: An example ................................. 335
References .......................................... 339
H. Hauser
Excellent surfaces and their taut resolution .................... .. 341
1 Introduction ........................................ . 341
2 Preliminaries ...................................... .. 346
3 Definition of the centers of blowup ........................ . 348
4 Transformation of equimultiple locus under blowup ............ . 353
5 Transformation of flags under blowup ...................... . 359
6 Construction of the induction invariant ..................... . 364
7 Transformation of iax under monomial blowup ............... . 367
8 Reduction to monomial blowup .......................... . 369
9 Proof of Theorem 1.1 ................................. . 371
References 372
A.J. de Jong
An application of alterations to Dieudonne modules .............. .. 375
Introduction ........................................ . 375
1 The application of alterations ............................ . 376
2 Full faithfulness up to isogeny ........................... . 376
3 Essential surjectivity up to isogeny ........................ . 378
References ......................................... . 380
F.-V. Kuhlmann
Valuation theoretic and model theoretic aspects
of local uniformization ..................................... 381
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 381
2 What does local uniformization mean? ...................... 381
3 Local uniformization and the Implicit Function Theorem .. . . . . . .. 388
4 Hensel's Lemma ...................................... 389
5 A crash course in ramification theory ....................... 394
6 A valuation theoretical interpretation of local uniformization ...... 399
7 Inertial generation and Abhyankar places .................... 401
Table of Contents ix
8 The defect .......................................... 404
9 Maximal immediate extensions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 408
10 A quick look at Puiseux series fields . . . . . . . . . . . . . . . . . . . . . . .. 410
11 The tame and the wild valuation theory ..................... 412
12 Some notions and tools from model theoretic algebra . . . . . . . . . . .. 415
13 Saturation and embedding lemmas . . . . . . . . . . . . . . . . . . . . . . . .. 421
14 The Generalized Grauert-Remmert Stability Theorem ........... 424
15 Relative local uniformization ...... . . . . . . . . . . . . . . . . . . . . . .. 428
16 Local uniformization for Abhyankar places ................... 432
17 Non-Abhyankar places and the Henselian Rationality
of immediate function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 433
18 Bad places ................ . . . . . . . . . . . . . . . . . . . . . . . . .. 437
19 The role of the transcendence basis and the dimension . . . . . . . . . .. 439
20 The space of all places of FIK .. . . . . . . . . . . . . . . . . . . . . . . . . .. 443
21 lFp((t)) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 446
22 Local uniformization vs. Ax-Kochen-Ershov .................. 448
23 Back to local uniformization in positive characteristic ........... 451
References .......................................... 452
D.T. Le
Les singularites Sandwich ................................... 457
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 457
2 Normalisation et eclatements . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 458
3 Singularites Sandwich ........ . . . . . . . . . . . . . . . . . . . . . . . . .. 460
4 Approche combinatoire ................................. 462
5 Singularites primitives, singularites minimales . . . . . . . . . . . . . . . .. 472
6 Courbes polaires des singularites rationnelles . . . . . . . . . . . . . . . . .. 475
7 Transformation de Nash des singularites minimales ............. 477
8 La resolution des singularites des surfaces via
la transformee de Nash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 479
References .......................................... 481
J. Lipman
Equisingularity and simultaneous resolution of singularities .......... 485
1 Introduction - equisingular stratifications .................... 485
2 Stratifying conditions .................................. 488
3 The Zariski stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 494
4 Equiresolvable stratifications ............................. 498
5 Simultaneous resolution of quasi-ordinary singularities . . . . . . . . . .. 501
References .......................................... 503