SINGULARITIES

http://dx.doi.org/10.1090/pspum/040.2

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

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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- I l l surface singularities defined by homo­geneous 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 P 2

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 singu­larities

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 O F 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 M A C P H E R S O N

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 Faa' de Bruno's formula 423 for higher derivatives of composite functions and the expression of some intrinsic derivatives

FELICE RONGA

Le type topologique eclate d'une 433 application analytique

C SABBAH

The higher residue pairings K{F

k) for a 441 family of hypersurface singular points

KYOJI SAITO

On the exponents and the geometric genus 465 of an isolated hypersurface singularity

MORIHIKO SAITO

A note on two local Hodge titrations 473 JOHN SCHERK

XIV TABLE OF CONTENTS

A cobordism invariant for surface 479 singularities

JOSE A. SEADE

Isolated line singularities 485 DIRK SIERSMA

A convexity theorem 497 ANDREW JOHN SOMMESE

Triple contact of plane curves: 507 Schubert's enumerative theory

ROBERT SPEISER

Mixed Hodge structures associated with 513 isolated singularities

J. H. M. STEENBRINK

The tangent space and exponential map 537 at an isolated singular point

DAVID A. STONE

Singularities of complex analytic 551 foliations

TATSUO SUWA

The exponents of a free hypersurface 561 HIROAKI TERAO

Local triviality of families of maps 567 J. G. TIMOURIAN

Comparing regularity conditions on 575 stratifications

DAVID TROTMAN

On singularities on degenerate Del Pezzo 587 surfaces of degree 1, 2

TOHSUKE URABE

The structure of quasihomogeneous 593 singularities

PHILIP WAGREICH

Derivations, automorphisms and deformations 613 of quasi-homogeneous singularities

JONATHAN M. WAHL

Classification of unimodal isolated 625 singularities of complete intersections

C. T. C. WALL

TABLE O F CONTENTS XV

^-functions and exponents of hypersurface 641 isolated singularities

TAMAKI YANO

On irregularity and geometric genus of 653 isolated singularities

STEPHEN S.-T. YAU

The structure of strata ju = const in a 663 critical set of a complete intersection singularity

YOSEF YOMDIN

Some results of finite determinacy and 667 stability not requiring the explicit use of smoothness

YOSEF YOMDIN

L2-cohomology and intersection homology 675 of locally symmetric varieties

STEVEN ZUCKER