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PROCEEDING S O F SYMPOSI A IN PUR E MATHEMATIC S
Volume 46, Part 2
Algebrai c Geometr y Bowdoin198 5
Spence r J. Bloch , Edito r
with the collaboratio n of H . Clemens , D. Eisenbud , W . Fulton , D. Gieseker , J. Harris ,
R. Hartshorne , an d S. Mor i
AMERICA N MATHEMATICA L SOCIET Y
PROVIDENCE , RHOD E ISLAN D
http://dx.doi.org/10.1090/pspum/046.2
PROCEEDINGS OF THE SUMMER RESEARCH INSTITUTE ON ALGEBRAIC GEOMETRY
HELD AT BOWDOIN COLLEGE BRUNSWICK, MAINE
JULY 8-26 , 1985
with support from the National Science Foundation, Grant DMS-8415200
1980 Mathematics Subject Classification (1985 Revision): Primary 14-06, 14-XX, 11G25, 11L40, 11Q25, 11R39, 13CXX, 13D05, 13E05, 13H15,
32JXX, 53C55. Secondary 11G40, 13DXX, 13E15, 13F20, 13H10, 20C99, 22E55, 30F30, 32G13,
43A32, 55P62, 57D35.
Library of Congress Cataloging-in-Publication Data
Summer Research Institute on Algebraic Geometry (1985: Bowdoin College) Algebraic geometry: Bowdoin 1985/Spencer J. Bloch, editor; with the collaboration
of H. Clemens. . . [et al.].
(Proceedings of symposia in pure mathematics, 0082- 0717; v. 46) Proceedings of Summer Research Institute on Algebraic Geometry, Bowdoin College,
Brunswick, Maine, July 8 -26 , 1985. Includes bibliographies. 1. Geometry, Algebraic—Congresses. I. Bloch, Spencer. II. Clemens, C. Herbert (Charles
Herbert), 1939- . III. American Mathematical Society. IV. Title. V. Series. QA564.S86 1985 516.3'5 87-12306 ISBN 0-8218-1476-1 (part 1) ISBN 0-8218-1480-X (part 2) ISBN 0-8218-1481-8 (set) (alk. paper)
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Contents PARTI
Algebraic Varieties; General Results
On varieties of minimal degree DAVID ElSENBUD AND JOE HARRIS 3
On the topology of algebraic varieties WILLIAM FULTON 15
Curves
Fay's trisecant formula and a characterization of Jacobian varieties ENRICO ARBARELLO
Deformations and smoothing of complete linear systems on reducible curves M E I - C H U CHANG AND ZIV R A N
Complete subvarieties of the moduli space of smooth curves STEVEN DIAZ
The irreducibility of the Hilbert scheme of smooth space curves LAWRENCE E I N
On theta functions for Jacobi varieties R. C. GUNNING
Curves and their moduli J O E HARRIS
On the classification of algebraic space curves, II ROBIN HARTSHORNE
The rationality of certain spaces associated to trigonal curves N. I. SHEPHERD-BARRON
Surfaces
Canonical rings and "special" surfaces of general type F . CATANESE
An introduction to the geography of surfaces of general type U L F PERSSON
Threefolds
Contributions to Riemann-Roch on projective 3-folds with only canonical singularities and applications
A. R. FLETCHER 221
iii
49
63
77
83
89
99
145
165
175
195
IV CONTENTS
Vanishing theorems for cohomology groups JANOS KOLLAR 233
Deformation of a morphism along a foliation YOICHI MlYAOKA 245
Classification of higher-dimensional varieties SHIGEFUMI M O R I 269
Tendencious survey of 3-folds MILES R E I D 333
Young person's guide to canonical singularities MILES R E I D 345
Affine Algebraic Geometry
Classification of noncomplete algebraic varieties TAKAO FUJITA 417
The Zariski decomposition of log-canonical divisors YUJIRO KAWAMATA 425
Open algebraic surfaces with Kodaira dimension -oo M. MlYANISHI AND S. TSUNODA 435
Classification of normal surfaces FUMIO SAKAI 451
Divisors with finite local fundamental group on a surface A. R. SHASTRI 467
PART 2
Groups in Algebraic Geometry
Rationality of fields of invariants IGOR V. DOLGACHEV 3
Automorphisms of K3-like rational surfaces BRIAN HARBOURNE 17
Fundamental groups of the complements to plane singular curves A. LIBGOBER 29
Galois coverings in the theory of algebraic surfaces B . MOISHEZON AND M. TEICHER 47
Vector Bundles
Geometry of the Horrocks-Mumford bundle KLAUS HULEK 69
Variet£s de modules de faisceaux semi-stables de rang eleve sur P2 J . LE POTIER 87
Vector bundles and submanifolds of projective space: Nine open problems MICHAEL SCHNEIDER 101
CONTENTS v
Geometry in Characteristic p
F-isocrystals and p-adic representations RICHARD C R E W i l l
Foliations and inseparable morphisms TORSTEN EKEDAHL 139
Deligne's £-adic Fourier transform LUC ILLUSIE 151
Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero
FRANS OORT 165
Hodge Theory
The geometry of the extension class of a mixed Hodge structure JAMES A. CARLSON 199
The local geometry of the Abel-Jacobi mapping HERBERT CLEMENS 223
Generic Torelli and infinitesimal variation of Hodge structure DAVID A. COX 235
The geometry of the mixed Hodge structure on the fundamental group RICHARD M. HAIN 247
Degeneration of mixed Hodge structures STEVEN ZUCKER 283
Enumerative Geometry
Hilbert scheme of points: Overview of last ten years A. IARROBINO 297
Intersection theory and enumerative geometry: A decade in review STEVEN L. KLEIMAN WITH ANDERS THORUP 321
Completed quadrics and linear maps DAN LAKSOV 371
Local Chern characters and intersection multiplicities PAUL C. ROBERTS 389
Enumerating contacts ROBERT SPEISER 401
Algebraic Cycles
Cycles on arithmetic schemes and Euler characteristics of curves SPENCER BLOCH 421
Zero-cycles and if-theory on singular varieties MARC LEVINE 451
Zero cycles modulo rational equivalence for some varieties over fields of transcendence degree one
CHAD SCHOEN 463
vi CONTENTS
Rational equivalence of 0-cycles on normal varieties over C V. SRINIVAS
Commutative Algebra
Syzygies: The codimension of zeros of a nonzero section E. GRAHAM EVANS, JR . AND PHILLIP GRIFFITH
Intersection problems and Cohen-Macaulay modules MELVIN HOCHSTER
Decompositions of torsionfree modules over affine curves ROGER WIEGAND AND SYLVIA WIEGAND