Table Of ContentProgress 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
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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
