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SINGULARITIES PROCEEDINGS of SYMPOSIA in PURE MATHEMATICS Volume 40, Part 2 http://dx.doi.org/10.1090/pspum/040.2 SINGULARITIES PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 40, Part 2 SINGULARITIES AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 40 PROCEEDINGS OF THE SUMMER INSTITUTE ON SINGULARITIES HELD AT HUMBOLDT STATE UNIVERSITY ARCATA, CALIFORNIA JULY 20-AUGUST 7, 1981 EDITED BY PETER ORLIK Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 80-20208 1980 Mathematics Subject Classification. Primary 14-XX, 16-XX, 32-XX, 53-XX, 55-XX, 57-XX, 58-XX. Library of Congress Cataloging in Publication Data Main entry under title: Singularities. (Proceedings of symposia in pure mathematics; v. 40, pts. 1, 2) Bibliography: p. 1. Singularities (Mathematics)-Addresses, essays, lectures. I. Orlik, Peter, 1938- . II. Series. QA614.58.S55 1983 514'.74 83-2529 ISBN 0-8218-1443-5 (set) ISBN 0^8218-14664 (part 2) ISBN 03218-1450-8 (part 1) ISSN 0082-0717 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit libraries acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under Ucense from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, Box 6248, Providence, Rhode Island 02940. The appearance of the code on the first page of an article in this journal indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U. S. Copyright Law, provided that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright © 1983 by the American Mathematical Society. Printed in the United States of America. All rights reserved except those granted to the United States Government. TABLE OF CONTENTS PART 1 Preface xx List of participants Desingularization of plane curves 1 SHREERAM S. ABHYANKAR Zeros of holomorphic vector fields 47 and the Gysin homomorphism E. AKYILDIZ AND J. B. CARRELL On the graded rings associated to 55 holomorphic vector fields with exactly one zero E. AKYILDIZ, J. B. CARRELL, D. I. LIEBERMAN AND A. J. SOMMESE Some open problems in the theory 57 of singularities V. I. ARNOLD Complex surfaces of small homotopy 71 type GOTTFRIED BARTHEL Homeomorphy classification of normal 81 surfaces in P3 with C*-action GOTTFRIED BARTHEL Minimal free resolutions of Gorenstein 105 local rings with small multiplicity KURT BEHNKE Analytic equivalence of isolated hyper- Ill surface singularities defined by homo• geneous polynomials MAX BENSON v VI TABLE OF CONTENTS An application of singularity theory to 119 nonlinear elliptic partial differential equations M. S. BERGER, P. T. CHURCH AND J. G. TIMOURIAN Algebras of composite differentiable 127 functions EDWARD BIERSTONE AND PIERRE D. MILMAN The extension problem and related 137 themes in differential analysis EDWARD BIERSTONE AND GERALD W. SCHWARZ On global extensions of Dynkin diagrams 145 and singular surfaces of the topological type of P2 LAWRENCE BRENTON, DAVID BINDSCHADLER, DANIEL DRUCKER AND GEERT C. E. PRINS Milnor lattices and Dynkin diagrams 153 E. BRIESKORN On the topology of polynomial hyper- 167 surfaces S. A. BROUGHTON Genericity of caustics by reflexion 179 J. W. BRUCE, P. J. GIBLIN AND C. G. GIBSON Envelopes, duality and contact structures 195 J. W. BRUCE Bifurcation involving the hexagonal 203 lattice ERNESTO BUZANO AND MARTIN GOLUBITSKY On saturations of curve singularities 211 (any characteristic) ANTONIO CAMPILLO A note on the classes [£,*(/)] 221 DANIEL S. CHESS Differential maps with small 225 critical set P. T. CHURCH The unfolding and determinacy theorems 233 for subgroups of & and % JAMES DAMON TABLE OF CONTENTS Vll Topological triviality in versal unfoldings 255 JAMES DAMON Newton titrations, monomial algebras 267 and nonisolated and equivariant singu• larities JAMES DAMON Contact germs from the plane to the plane 277 A. DIMCA AND C. G. GIBSON Weyl groups and Cremona transformations 283 IGOR V. DOLGACHEV Genericity and smooth finite determinacy 295 ANDREW DU PLESSIS A naive guide to mixed Hodge theory 313 ALAN H. DURFEE The low-dimensional topology of 321 singularities ALAN H. DURFEE On the monodromy groups of singularities 327 WOLFGANG EBELING Rational curves with cusps 337 DAVID EISENBUD Mixed Hodge structures 345 FOUAD ELZEIN Special polars and curves with one 353 place at infinity ROBERT EPHRAIM Another view of critical point theory 361 ISTVAN FARY Some results on Poincare duality 373 ISTVAN FARY Some essential valuations in the resolution 381 of singularities JONATHAN FINE Derivations and smooth discriminant 385 KLAUS FISCHER Vlll TABLE OF CONTENTS Singularities of period maps and the 391 weak global Torelli problem ROBERT FRIEDMAN AND ROY SMITH The Thorn polynomial of 2TTTT 399 TERENCE GAFFNEY The structure of T&(f), classification 409 and an application to differential geometry TERENCE GAFFNEY Multiple points and associated ramification 429 loci TERENCE GAFFNEY Equivalence theorems in global singularity theory 439 TERENCE GAFFNEY AND LESLIE WILSON Hilbert scheme as flattener 449 A. GALLIGO Differential invariance of multiplicity 453 YIH-NAN GAU AND JOSEPH LIPMAN Classification des singularites isolees 457 simples d'intersections completes MARC GIUSTI Finding the nondegenerate quadratic 495 singularities NORMAN GOLDSTEIN A discussion of symmetry and symmetry 499 breaking MARTIN GOLUBITSKY AND DAVID SCHAEFFER Stratified Morse theory 517 MARK GORESKY AND ROBERT MACPHERSON On the topology of smoothable singularities 535 GERT-MARTIN GREUEL AND JOSEPH STEENBRINK Lefschetz theorems for singular varieties 547 HELMUT A. HAMM Stratification via corank one projections 559 ROBERT M. HARDT An algorithm of construction of the 567 semiuniversal deformation of an isolated singularity HERWIG HAUSER TABLE OF CONTENTS IX Limites de normales, conditions de 575 Whitney et eclatement d'Hironaka J. P. G. HENRY ET M. MERLE Computation of some projective invariants 585 AUDUN HOLME Deforming complete intersection Artin 593 algebras. Appendix: Hilbert function of C[.x, y]/I ANTHONY IARROBINO Duality for Riemannian foliations 609 FRANZ W. KAMBER AND PHILIPPE TONDEUR Normally flat deformations of rational 619 and minimally elliptic singularities ULRICH KARRAS The topology of real algebraic sets 641 H. KING Differential forms and the fundamental 655 group of the complement of hypersurfaces TOSHITAKE KOHNO Questions about the proof of the hard 663 Lefschetz theorem KLAUS LAMOTKE Metric study of the neighbourhood of a 669 singularity REMI LANGEVIN TABLE OF CONTENTS PART 2 Weak simultaneous resolution for deforma- 1 tions of Gorenstein surface singularities HENRY B. LAUFER Introduction to linear differential systems 31 LE DUNG TRANG AND ZOGHMAN MEBKHOUT Cycles evanescents, sections planes et 65 conditions de Whitney. II LE D. T. ET B. TEISSIER Report on the problem session 105 LE D. T. AND B. TEISSIER Extended Artingroups 117 HARM VAN DER LEK Differentiable structures on complete 123 intersections. II ANATOLY S. LIBGOBER AND JOHN W. WOOD Alexander invariants of plane algebraic 135 curves A. LIBGOBER A connection between polar invariants and 145 roots of the Bernstein-Sato polynomial BEN LICHTIN Estimates and formulae 155 for the C° degree of sufficiency of plane curves BEN LICHTIN Quasi-ordinary singularities of 161 surfaces in C3 JOSEPH LIPMAN XI xu TABLE OF CONTENTS The smoothing components of a triangle 173 singularity. I EDUARD LOOIJENGA On the structure of embedded algebroid 185 surfaces IGNACIO LUENGO Some new surfaces of general type 193 RICHARD MANDELBAUM Distance from a submanifold in 199 Euclidean space JOHN N. MATHER On the topology of Deligne's weight 217 filtration CLINT McCRORY Topological types of 227 polynomial map germs ISAO NAKAI Abelian covers of quasihomogeneous 233 surface singularities WALTER D. NEUMANN Geometry of quasihomogeneous surfaces 245 singularities WALTER D. NEUMANN On the stability of the Newton boundary 259 MATSUO OKA Coxeter arrangements 269 PETER ORLIK AND LOUIS SOLOMON Vanishing folds in families of singularities 293 DONAL B. O'SHEA Projective resolutions of Hodge algebras: Some examples 305 JURGEN PESSELHOY AND OSWALD RIEMENSCHNEIDER Vanishing homologies and the n variable 319 saddlepoint method FREDERIC PHAM A note on higher order dual varieties, 335 with an application to scrolls RAGNI PIENE TABLE OF CONTENTS Xlll Factorization of birational maps in 343 dimension 3 HENRY C. PINKHAM Smoothings of the Dpqr singularities, 373 p + q + r = 22 HENRY C. PINKHAM The normal singularities of surfaces in R3 379 IAN R. PORTEOUS Probing singularities 395 IAN. R. PORTEOUS C*-equivariant deformations of germs of 407 coherent sheaves FERNANDO PUERTA The real Jacobian problem 411 JOHN D. RANDALL Milnor fibers and Alexander polynomials 415 of plane curves RICHARD RANDELL Multiple points of real mappings 421 FELICE RONGA A new look at
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