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 60 th 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