Real and Complex Singularities (Lecture Notes in Pure and Applied Mathematics)

Real and Complex Singularities the sixth workshop at Sao Carlos

edited by

David Mond University of Warwick Coventry, England

Marcela Jose 5aia University of Sao Paulo Sao Carlos, Sao Paulo, Brazil

MARCEL DEKKER, INC.

NEW YORK • BASEL

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN: 0-8247-4091-2 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-260-6300; fax: 41-61-260-6333 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above.

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PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

Zuhair Nashed University of Delaware Newark, Delaware

EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitdt Siegen

W. S. Massey Yale University

Mark Teply University of Wisconsin, Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

N. Jacobson, Exceptional Lie Algebras L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et a/., Differentiable Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characten'zation of C(X) Among Its Subalgebras 6. R. McDonald et a/., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et a/., Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin etal., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L Cesari et a/., Nonlinear Functional Analysis and Differential Equations J. J. Schaffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et a/., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et a/., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory B. Kadem, Binary Time Series J. Barros-Neto_and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L Stemberg et a/.. Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et a/., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory

60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.

J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A. Me/son, Direct Integral Theory J. E. Smith et a/., Ordered Groups J. Cronin, Mathematics of Cell Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series 7. G. McLaughlin, Regressive Sets and the Theory of Isols T. L Herdman et a/., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Gee/, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A. Fiacco, Mathematical Programming with Data Perturbations I P. Schultz et a/., Algebraic Structures and Applications L Bican et a/., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et a/., Volterra and Functional Differential Equations N. L. Johnson era/., Finite Geometries G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-8. Hin'art-Umityetal., Optimization A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istratescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L P. de Alcantara, Mathematical Logic and Formal Systems C. E. Aull, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L. Salce, Modules Over Valuation Domains P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightboume III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations C. A. Baker and L. M. Batten, Finite Geometries J. W. Brewer et a/., Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology D. W. Kueke et a/., Mathematical Logic and Theoretical Computer Science B.-L. Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M. C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson et a/.. Transport Theory, Invariant Imbedding, and Integral Equations P. Clement et a/., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et a/., Differential Equations E. O. Roxin, Modem Optimal Control J. C. Diaz, Mathematics for Large Scale Computing

121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.

P. S. Milojevft Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics W. D. Wallis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Parameter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Anno ef a/., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et a/., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et a/., General Topology and Applications P Clement et al., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et al., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et al., Methods in Module Theory G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Joshi and A. V. Balakhshnan, Mathematical Theory of Control G. Komatsu and Y, Sakane, Complex Geometry /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections L. Fuchs and R. Gobel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et al., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt et al., Functional Analysis K. G. Fischer et al., Computational Algebra K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et al., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Brayetal., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolesio, Boundary Control and Variation M. Kfizeketal., Finite Element Methods G. Da Prafo and L. Tubaro, Control of Partial Differential Equations £ Ballico, Projective Geometry with Applications M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et al., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini etal. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Agliano, Logic and Algebra X. H. Cao et al., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

184. 185. 186. 187. 188.

J. E. Andersen et al., Geometry and Physics P.-J. Cahen et a/., Commutative Ring Theory J. A. Goldstein et a/., Stochastic Processes and Functional Analysis A. Sorbi, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis 189. D. D. Anderson, Factorization in Integral Domains 190. N. L. Johnson, Mostly Finite Geometries 191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems 192. W. H. Schikhofet a/., p-adic Functional Analysis 193. S. Sertoz, Algebraic Geometry 194. G. Caristi and E. Mitidieri, Reaction Diffusion Systems 195. A. V. Fiacco, Mathematical Programming with Data Perturbations 196. M. Kfizek et a/., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates 197. S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups 198. V. Drensky et al., Methods in Ring Theory 199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions 200. P. E. Newstead, Algebraic Geometry 201. D. Dikranjan and L Salce, Abelian Groups, Module Theory, and Topology 202. Z. Chen et al., Advances in Computational Mathematics 203. X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs 204. C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications 205. D. E. Dobbs et al., Advances in Commutative Ring Theory 206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry 207. J. Kakol et al., p-adic Functional Analysis 208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory 209. S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups 210. F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras 211. R. Costa et al., Nonassociative Algebra and Its Applications 212. T.-X. He, Wavelet Analysis and Multiresolution Methods 213. H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference 214. J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis 215. G. Lumerand L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences 216. J. Cagnol et al., Shape Optimization and Optimal Design 217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra 218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems 219. F. AH Mehmeti et al., Partial Differential Equations on Multistructures 220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra 221. A. Gran/a et al., Ring Theory and Algebraic Geometry 222. A. K. Katsaras et al., p-adic Functional Analysis 223. R. Salvi, The Navier-Stokes Equations 224. F. U. Coelho and H. A. Merklen, Representations of Algebras 225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory 226. G. Lyubeznik, Local Cohomology and Its Applications 227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications 228. W. A. Carnielli et al., Paraconsistency 229. A. Benkirane and A. Touzani, Partial Differential Equations 230. A. Illanes et al., Continuum Theory 231. M. Fontana et al., Commutative Ring Theory and Applications 232. D. Mond and M. J. Saia, Real and Complex Singularities 233. V. Ancona and J. Vaillant, Hyperbolic Differential Operators Additional Volumes in Preparation

Preface The Workshop on Real and Complex Singularities is a series, initiated in 1990 by Maria A. S. Ruas, of biennial workshops organized by the Singularity Theory group at the Institute of Mathematical Sciences and Computation, of the University of Sao Paulo, Sao Carlos (ICMC-USP, Sao Carlos), Brazil. Its main purpose is to bring together specialists in singularity theory and related fields. This book contains papers presented at the 6th Workshop on Real and Complex Singularities. It focuses on the role of singularity theory in algebraic geometry, mathematical physics, differential geometry, and dynamical systems. The meeting consisted of 13 plenary sections and 30 sections divided into three categories. The first deals with algebro-geometric aspects of singularity theory, the second is dedicated to singularity theory itself and the third is concerned with applications of singularity theory to mathematical physics, dynamical systems, and geometry. Three mini-courses were given also, one by Abramo Hefez, another by Terry Gaffney and Steven Kleiman, and one by Glaus Hertling. The papers presented here are a selection of those submitted to the editors. They are grouped into three categories: the first consists of the notes of the mini-course Irreducible Plane Curve Singularities by Abramo Hefez. This is in fact a small book on this subject, covering the following topics: rings of power series and Hensel's Lemma; the Preparation Theorem; the Hilbert-Riickert Basis Theorem; algebroid plane curves; Newton-Puiseux Theorem; plane analytic curves; intersection of curves; the semigroup of values; Apery sequences; resolution of singularities; Max Noether's formula; Milnor's number and the conductor; contact among two plane branches; intersection indices of two curves. The second group, dedicated to singularity theory, starts with the paper of C.T.C. Wall on Openness and multitransversality, followed by the paper of D. Barlet and A. Jeddi, The distribution fA fs D and the real asymptotic spectrum, and the papers of I. Scherbak, Deformations of boundary singularities and non-crystallographic Coxeter groups, S. Izumiya and K. Maruyama, Transversal Whitney topology and singularities of Haefliger foliations, V. S. Kulikov, On a conjecture of Chisini for coverings of the plane with A-D-E-singularities, R. G. W. Atique and D. Mond, Not all codimension 1 germs have good real pictures, J. Seade, On the topology of hypersurface singularities, V. H. J. Perez, Polar multiplicities and equisingularity of map germs from C3 to C 4 , D. Hacon, C. Mendez de Jesus, and M. C. Romero Fuster, Topological invariants of stable maps from a surface to the plane from a global viewpoint and R. Bulajich, L. Kushner and S. Lopez de Medrano, Cubics in R and C. The third category includes applications of singularity theory to mathematical physics, dynamical systems, and differential geometry. We have the articles Indices of Newton non-degenerate vector fields and a conjecture of Loewner for surfaces in R 4 , by C. Gutierrez and M. A. S. Ruas, Generic singularities of H-directions by L. F. Mello, Vertices of curves on constant curvature manifolds by S. R. Costa and C. C. Pansonato, and Projections of hypersurfaces in R4 to planes by A. C. Nabarro.

iv

Preface

Last, but not least, we have the notes of Glaus Hertling's mini-course Frobenius manifolds and hypersurface singularities. This introduces the general theory of Frobenius manifolds, and then shows how to endow the base-space of a miniversal deformation of a hypersurface singularity with the structure of a Frobenius manifold. As an application, the author constructs global moduli spaces for isolated hypersurface singularities. Thanks are due to many people and institutions. We start by thanking the members of the organizing committee, Angela Sitta, Miriam Manoel, and Ton Marar and the members of the scientific committee: Takuo Fukuda, Terry Gaffney, Carlos Gutierrez, Steven Kleiman. and Alexandre Varchenko for their support. We also thank Maria Ruas and all the staff of the ICMC. Without their help we could not have organized the workshop. The workshop was funded by FAPESP, CNPq, CAPES, USP, and SBM, whose support we gratefully acknowledge. It is a pleasure to thank the speakers and the participants whose presence was the real success of the Qth Workshop. We thank the staff members of Marcel Dekker, Inc., involved with the preparation of this book, and all those who have contributed in whatever way to these proceedings. All the papers here have been refereed. David Mond Marcelo Jose Saia

Contents Preface Contributors

Hi vii

1.

Irreducible Plane Curve Singularities Abramo Hefez

2.

Openness and Multitransversality C. T. C. Wall

3.

The Distribution fA fs n and the Real Asymptotic Spectrum Daniel Barlet and Ahmed Jeddi

137

4.

Deformations of Boundary Singularities and Non-Crystallographic Coxeter Groups Ina Scherbak

151

5. 6.

1 '

Transversal Whitney Topology and Singularities of Haefliger Foliations Shyuchi Izumiya and Kunihide Maruyama . On a Conjecture of Chisini for Coverings of the Plane with A-D-E-Singularities Valentine S. Kulikov

121

165

175

I.

Not All Codimension 1 Germs Have Good Real Pictures David Mond and Roberta G. Wik Atique

189

8.

On the Topology of Hypersurface Singularities Jose Seade

201

9.

Polar Multiplicities and Equisingularity of Map Germs from C3 to C4 Victor Hugo Jorge Perez

207

10.

Topological Invariants of Stable Maps from a Surface to the Plane from a Global Viewpoint D. Hacon, C. Mendez de Jesus, and M. C. Romero Fuster

227

II.

Cubics in R and C Radmila Bulajich, Leon Kushner, and Santiago Lopez de Medrano

12.

Indices of Newton Non-Degenerate Vector Fields and a Conjecture of Loewner for Surfaces in R4 Carlo Gutierrez and Maria Aparecida Soares Ruas

13.

Generic Singularities of H-Directions Luis Fernando O. Mello

237

245 255

vi

Contents

14.

Vertices of Curves on Constant Curvature Manifolds Claudia C. Pansonato and Sueli I. R. Costa

267

15.

Projections of Hypersurfaces in R4 to Planes Ana Claudia Nabarro

283

16.

Frobenius Manifolds and Hypersurface Singularities Claus Hertling

301

Contributors Daniel Barlet Universite Henri Poincare and Institut Elie Cartan, Vandoeuvre-lesNancy, France Radmila Bulajich Sueli I. R. Costa

Universidad Autonoma del Edo. de Morelos, Morelos, Mexico Institute di Matematica - Unicamp, Campinas - SP, Brazil

Carlos Gutierrez Vidalon

Universidade de Sao Paulo, Sao Carlos, SP, Brazil

D. Hacon PUC-RIO, Gavea - Rio de Janeiro, Brazil Abramo Hefez

Universidade Federal Fluminense, Niteroi, RJ, Brazil

Claus Hertling

Universitat Bonn, Bonn, Germany

Shyuichi Izumiya

Hokkaido University, Sapporo, Japan

Ahmed Jeddi Universite Henri Poincare and Institut Elie Cartan, Vandoeuvre-lesNancy, France Valentine S. Kulikov

Moscow State University of Printing, Moscow, Russia

Leon Kushner Universidad Nacional Autonoma de Mexico, Mexico DF, Mexico Santiago Lopez de Medrano DF, Mexico Kunihide Maruyama*

Hokkaido University, Sapporo, Japan

Luis Fernando O. Mello C. Mendes de Jesus David Mond

Universidad Nacional Autonoma de Mexico, Mexico

Escola Federal de Engenharia de Itajuba, Itajuba, MG, Brazil

PUC-RIO, Gavea - Rio de Janeiro, Brazil

University of Warwick, Coventry, United Kingdom

Ana Claudia Nabarro

Universidade de Sao Paulo, Sao Carlos, SP, Brazil

Claudia C. Pansonato

CCNE - UFSM, Santa Maria-RS, Brazil

Victor Hugo Jorge Perez

Universidade Estadual de Maringa, Maringa (PR), Brazil

^Current affiliation: NEC Nogawaryou, Kawasaki, Japan.

viii

Contributors

M. C. Romero Fuster Ina Scherbak Jose Seade

Universitat de Valencia, Burjasot (Valencia), Spain

Tel Aviv University, Ramat Aviv, Israel

Universidad Nacional Autonoma de Mexico DF, Morelos, Mexico

Maria Aparecida Scares Ruas C. T. C. Wall

Universidade de Sao Paulo, Sao Carlos, SP, Brazil

The University of Liverpool, Liverpool, United Kingdom

Roberta G. Wik Atique

Universidade de Sao Paulo, Sao Carlos, SP, Brazil

Real and Complex Singularities

Irreducible Plane Curve Singularities ABRAMO HEFEZ Institute de Matematica, Universidade Federal Fluminense, R. Mario Santos Braga s/n, 24020-140, Niteroi, R.J. Brasil. E-mail: [email protected]

To my parents from whom I learned the essence of life.

INTRODUCTION The main objective of these notes is to introduce the reader to the local study of singularities of plane curves from an algebraic point of view. This small book is a kind oiformulaire where the working singularist can find the basic facts and formulas, with their complete proofs, that exist in this context. The subject, singularities of curves, has motivated for more than a century innumerable research work and is still a fertile field of investigation. To motivate the framework in which we will place ourselves, suppose that a nonconstant polynomial f ( X , Y ) £(E[X,Y] is given and consider the algebraic complex plane curve C = CV = {(*, n, which implies that (/i n ) n eiN is a Cauchy sequence, hence convergent to an element of H. The limit will be denoted by EAGA/A(ii) Given n 6 IX, we have that the set AJ = (A e A; mult (A/A + Bgx) < n}, is finite because there are finitely many A's satisfying the following inequality min{mult(A) + mult (/ A ), mult (B) + mult(_g A )} < mult ( A/A + Bgx} < n. Now, let A n be as above and put A; = {A e A; mult(0A) < n] . Defining 0n = EAeA;; WA +B9\) and V'n = ^EAeA r , /A +B EACA; #A, we have that 7/;n — 0n E Ai^+1, hence limn—^oo (i]>n — 0 n ) ~ 0, and the result follows considering the following equalities:

\—^ q\ lim ?/;n = A x—^ y fx + tiy

n—*oo

^—'

AeA

^—' '

and

AeA

N ilim o = \— / > (Af\ + Bg\). n n—>oo ^—' AeA

(iii) Let An and A^ be as above, and define A" = {(A,/y.) e A x A; mult(/ A ^) < n}. This last set is obviously finite. Now, defining E

' V^

"i

we have as above,

lim

n —>oo

which together with lim ^n - (V/ A )(V. 9 A ),

n—>oo

•*•—' AeA

^—' AeA

and lim $n = n—>oo

V

*•—' (A,^)eAxA

Irreducible Plane Curve Singularities

11

imply the result.

D

Let {Pi~ i e IN} be a family of homogeneous polynomials, with Pi e K[Yi, . . . Ys] of degree z, and let g±, . . . ,gs € Mn- the family

is summable, because of the inequalities (see Problem 1.3) l,...,^))>z,

V i e IN.

If / = EigjN P»(yi' • • • > ys) E ^[[yi' • • • » y*]]' then the sum of the family .F will be denoted by / (g\ , . . . , gs ) , and will be called the substitution of Yi , . . . , Ys by g± , . . . , gs in /. In particular, it is possible to substitute Y\, . . . , Yr by 0, . . . , 0 in /, getting PQ. The following result follows immediately from the Proposition (1.8). COROLLARY 1.9 Given gi,...,gae Mn, f,he K [ [ Y i , . . . , Ya]] and a K[[Xlt...tXr]] » Sgi,...,g.(f) = f(9l,...,ga)

is a homomorphism of K -algebras . In the next section we will see that any homomorphism of /T-algebras is a substitution map. Notice that the condition #1, . . . , gs 6 M.-R. is essential in order to make substitutions. For example, if f ( X ) = l + X+X2 + - - - e K[[X]} , what would be /(I) ? More generally, given g\ , . . . , gs £*R,, such that for some z, QI (0) = c 6 K \ {0} , then taking /(Yi, . . . , Ys) = 1 + c~lYi + c~2Y^ -\ ---- , we have that f ( g i , . ..,gs] is not defined as an element of 7£, since otherwise we would have

which is a contradiction. Let Pi € K[Xi, . . . , -X"r], i € IN be a family of homogeneous polynomials with deg(,Pi) = i. For all (j1} . . . , >) e INr the family

is summable. The sum of the family Q is what we call the partial derivative of / = ]CieiN -^ °f orfier ji in Xi , etc., and of order jr in Xr, and is denoted by QJ\-\— \-jrf

12

Hefez

PROBLEMS 2.1) Suppose that / 6 7£ is such that /(O) = a ^ 0. Using the identity in Problem 1.1 a, show that

2.2) Let /,- 6 U, i 6 IX, be such that lim?- mult(/;) = oo. a) Show that (/,•) is a Cauchy sequence in 7£, whose limit is 0 e 7£. b) Show that the family J7 = {/,;; '/' £ IN} is summable. 2.3) Let T — {/,;; i 6 IX} be a summable family of power series in 7£. a) Show that the infinite sum Y^Li /' ^s independent of the order in which we write the elements /,. b) Show that mult(^~ 0 /i) > min{mult(/ i );i 6 IX}. 2.4) Show that with the metric we defined in 7?., we have: a) The operations of addition and multiplication are continuous. b) The substitution homomorphism S(Jl .....5s is continuous. 2.5) Show that K[Xi,. . . , X r ] is dense in K [ [ X i , . . . , X r ] ] . Show also that K [ [ X \ , . . ., Xr_i}} is closed in K [ [ X \ , . . . ,Xr]]. 2.6) a) Show that if / € #[[Yi, - - - , Ya}} and that mult(^j) > 1, for i = 1, . . . , «, then

Sl, ...,ga

e K^X^ . . . , Xr]}, are such

mult(59l.....5 ,(/)) = mult(/(. 9l , . . . ,. 9s )) > mult(/) • min{mult(.g,-);« € IX}. b) Show that / e /^[[X]] \ {0} and .9 e ^[[A^, . . . , Xr]} \ {0}, with mult(c/) > 1, then mult(/( 5 )) = mult(/) • mult(.9). 2.7) Show that if the characteristic of K is zero, then the multiplicity of / 6 7£\ {0} may be determined as follows: multf /)7 - miri '

1.3 HOMOMORPHISMS Let 7£ = ^[[Xi , . . . , Xr}] and 5 — /^[[li , . . - , Ys]] . Denoting respectively their maximal ideals by M.-R anci MS, we have the following result: PROPOSITION 1.10 Let T :S —> U be a homomorphism of K -algebras. Then

ii)

T is continuous.

in)

There exist gi, . . . , gs 6 M.TI such that T = Sgi,...tgfi .

Irreducible Plane Curve Singularities

13

Proof: (i) Let / e Ms and suppose that T(f) £ Mn- Then we may write T(f) = c + g, where c e K \ {0} and g € M-JI- It then follows that T(f -c)=g, where / — c is invertible in <S, while g is not invertible in 7£, which is a contradiction. (ii) It follows immediately from (i) that T(M$) C M^, which easily implies that T is continuous. (iii) From (i) we have that T(Yj) = gj € Mn, for j = 1, . . . , s. Now let / = £ielN Pi. Since i=0

i=0

hnin^oo £"=0 Pi = /, linin^oo £™=0 P*(#i' • • • ' 9s) = f ( d i , • • • ,9s) and T is continuous, the result follows. D So, given a homomorphism T : S —> 7£, there exist gi,...,ga € M-R such that T = Sgij...:gs. Let us now see which additional conditions we must impose on #i, . . . , gs to guarantee that T is a K -isomorphism. Initially, observe that every /C-isomorphism from S to 7£ must preserve multiplicities. Indeed, from Problem 2.6(a), it follows that for all / G <S, mult(/) < mult(T(/)) < mult(T- 1 (T(/))) = mult(/), which proves our assertion. Since T(Yi) = #j, for i = 1, . . . , s, it follows that mult(^) = 1, for alH = 1, . . . , .s. Let L I , . . . , L S respectively the initial forms of g l5 . . . ,