Singularities: The Brieskorn Anniversary Volume

474 Pages·1998·14.27 MB·English

by  Yuri A. Drozd

Progress in Mathematics
Volume 162
Series Editors
H. Bass
J. Oesterle
A. Weinstein
Singularities
The Brieskom Anniversary Volume
Y.I.Arnold
G.-M. Greuel
J.H.M. Steenbrink
Editors
Springer Base1 AG
Editors:
V.I.Amold G.-M. GreueI
Department of Geometry and Topology Fachbereich Mathematik
Steklov Mathematical Institute Universitiit Kaiserslautem
8, Gubkina Stree D-67653 Kaiserslautem
117966 Moscow GSP-I Germany
Russia
and
CEREMADE J.H.M. Steenbrink
Universite Paris-Dauphine Subfaculteit Wiskunde
Place du Marechal de Lattre de Tassigny Katholieke Universiteit Nijmegen
Postfach 3049 Toemooiveld
F-75775 Paris Cedex 16e NL-6525 ED Nijmegen
France The Netherlands
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
Singularities : the Brieskom anniversary volume / V. 1. Amold ...
ed.
(Progress in mathematics ; VoI. 162)
ISBN 978-3-0348-9767-9 ISBN 978-3-0348-8770-0 (eBook)
DOI 10.1007/978-3-0348-8770-0
This work is subject to copyright. Ali rights are reserved, whether the whole or part of
the material is concemed, specifically the rights of translation, reprinting, re-use of
illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in
data banks. For any kind ofuse whatsoever, permission from the copyright owner must
be obtained.
© 1998 Springer Basel AG
Origina1ly published by Birkhlluser Verlag, Basel, Switzerland in 1998
Printed on acid-free paper produced of chlorine-free pulp. TCF 00
ISBN 978-3-0348-9767-9
987654321
Dedicated to
Egbert Brieskorn
on the Occasion of
His 60th Birthday
Prof. Dr. Egbert Brieskorn
Contents
Preface ................................................................ Xlii
Gert-Martin Greuel
Aspects of Brieskorn's mathematical work XV
Publication list ........................................................ XXlll
Chapter 1: Classification and Invariants
Yuri A. Drozd and Gert-Martin Greuel
On Schappert's Characterization of Strictly Unimodal
Plane Curve Singularities .............................................. 3
Introduction ....................................................... 3
1 Preliminaries... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Main theorem .................................................. 6
3 Ideals of ideal-unimodal plane curve singularities. . . . . . . . . . . . . . . . 17
References ......................................................... 26
Gert-Martin Greuel and Gerhard Pfister
Geometric Quotients of Unipotent Group Actions II 27
Introduction ....................................................... 27
1 Special representations .......................................... 29
2 Free actions .................................................... 32
References ......................................................... 36
Helmut A. Hamm
Hodge Numbers for Isolated Singularities of Non-degenerate
Complete Intersections ................................................ 37
Introduction ....................................................... 37
1 Mixed Hodge numbers for the link and
the vanishing cohomology ....................................... 38
2 Nondegenerate complete intersections ........................... 48
References ......................................................... 59
vii
viii Contents
Weiming Huang and Joseph Lipman
Differential Invariants of Embeddings of Manifolds
in Complex Spaces .................................................... 61
1 Normal cones................................................... 63
2 Specialization to the normal cone ............................... 67
3 Differential functoriality of the specialization over lR ............ 70
4 Multiplicities of components of C(V, W) ....................... 73
5 Relative complexification of the normal cone .................... 78
6 Segre classes .................................................... 87
References ......................................................... 92
Andras Nemethi
On the Spectrum of Curve Singularities 93
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2 The properties P(N) (2::::: N ::::: (0). ............................. 94
3 Positive results. The case of plane curve singularities ............ 96
4 The arithmetical approach revisited. Dedekind sums ............ 98
5 Germs without property P(N). ................................. 100
References ......................................................... 101
Mihai Tibiir
Embedding Nonisolated Singularities into Isolated Singularities 103
1 Introduction.................................................... 103
2 The main construction.......................................... 104
3 Homotopy type of the Milnor fibre .............................. 108
4 Zeta-function of the monodromy ................................ 111
References ......................................................... 114
Chapter 2: Deformation Theory
Andrew A. du Plessis and Charles T. C. Wall
Discriminants and Vector Fields....................................... 119
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 120
2 General theory of the discriminant .............................. 123
3 Construction of discriminant matrices and vector fields. . . . . . . . .. 126
4 Vector fields on the target of stable maps ....................... 127
5 Vector fields in the source ....................................... 129
6 The relation between vector fields in source and target .......... 134
7 The instability locus and the discriminant matrix ............... 135
References ......................................................... 138
Contents ix
Wolfgang Ebeling and Sabir M. Gusein-Zade
Suspensions of Fat Points and Their Intersection Forms 141
Introduction ....................................................... 141
1 p-fold suspensions of icis ....................................... 142
2 Convenient equations and the corresponding real picture ........ 143
3 A distinguished set of vanishing cycles
for the icis {x + zP = 0, x ± y2 = O} ............................ 145
4 Enumeration of vanishing cycles and the definition of
their orientations ............................................... 146
5 The equivariant intersection form. The equivariant
Picard-Lefschetz formula for the Ap- 1 singularity ............... 150
6 The intersection form of the suspension ......................... 152
7 Sketch of the proof of Theorem 2 ............................... 155
8 Relations between vanishing cycles of the distinguished set ...... 158
9 Examples....................................................... 159
References ......................................................... 165
Claus H ertling
Brieskorn Lattices and Torelli Type Theorems for Cubics in lP'3 and
for Brieskorn-Pham Singularities with Coprime Exponents ............ 167
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 167
2 Hypersurface singularities and polarized mixed Hodge structures 171
3 Brieskorn lattice ................................................ 175
4 The invariant BL ............................................... 180
5 Semiquasihomogeneous singularities with weights (~, ~, ~, ~) ... 183
6 Brieskorn-Pham singularities with pairwise coprime exponents .. 188
References ......................................................... 193
Eugenii Shustin
Equiclassical Deformation of Plane Algebraic Curves .................. 195
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
2 Preliminaries................................................... 198
3 Proof of Theorem 1.1 ........................................... 199
4 Proof of Corollary 1.3 ........................................... 202
5 Proof of Theorem 1.4 ........................................... 203
References ......................................................... 203
Victor A. Vassiliev
Monodromy of Complete Intersections and Surface Potentials ......... 205
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 205
2 Vanishing homology and local monodromy of
complete intersections .......................................... 207
3 Surface potentials and Newton-Ivory-Arnold theorem........... 211
x Contents
4 Monodromy group responsible for the ramification
of potentials .................................................... 214
5 Description of the small monodromy group and finiteness
theorems in the cases n = 2 and d = 2 .......................... 222
6 Proof of Theorems 7, 8 ......................................... 226
References ......................................................... 233
Appendix to the paper of V.A. Vassiliev (by Wolfgang Ebeling) ..... 235
References ......................................................... 237
Chapter 3: Resolution
Klaus Altmann
P-Resolutions of Cyclic Quotients from the Toric Viewpoint 241
1 Introduction... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 241
2 Cyclic quotient singularities ..................................... 242
3 The maximal resolution ......................................... 244
4 P-resolutions................................................... 246
References ......................................................... 249
Antonio Campillo and Gerard Gonzalez-Sprinberg
On Characteristic Cones, Clusters and Chains of
Infinitely Near Points ................................................. 251
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 251
2 Characteristic cones, complete ideals and
sandwiched varieties ........................................... . 252
3 Cones and constellations of infinitely near points ............... . 254
4 Clusters and chains of infinitely near points .................... . 258
References ........................................................ . 261
Heiko Cassens and Peter Slodowy
On Kleinian Singularities and Quivers 263
Introduction ...................................................... . 263
1 Reminder on Kleinian singularities ............................. . 264
2 McKay's observation ........................................... . 266
3 Symplectic geometry and momentum maps .................... . 267
4 Kronheimer's work ............................................. . 271
5 Quivers ........................................................ . 273
6 Linear modifications ........................................... . 277
7 Simultaneous resolution ........................................ . 280
References ........................................................ . 285