SINGULARITIES, Part 2 · On singularities on degenerate Del Pezzo 587 surfaces of degree 1, 2...
Transcript of SINGULARITIES, Part 2 · On singularities on degenerate Del Pezzo 587 surfaces of degree 1, 2...
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