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