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Algebraic Geometry: Sundance 1988 http://dx.doi.org/10.1090/conm/116 Titles in This Series

Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomaier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 . Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5,6, 7, 10, 11,12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, A. Frohlich 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R. Lidl, Editor and probability, Richard Beals, 10 Operator algebras and K-theory, Anatole Beck, Alexandra Bellow, and Ronald G. Douglas and Claude Arshag Hajian, Editors Schochet, Editors 27 Microlocal analysis, M. Salah Baouendi, 11 Plane ellipticity and related problems, Richard Beals, and Linda Preiss Robert P. Gilbert, Editor Rothschild, Editors 12 Symposium on algebraic topology in 28 Fluids and plasmas: geometry and honor of Jose Adem, Samuel Gitler, dynamics, Jerrold E. Marsden, Editor Editor 29 Automated theorem proving, W. W. 13 Algebraists' homage: Papers in Bledsoe and Donald Loveland, Editors ring theory and related topics, 30 Mathematical applications of category S. A. Amitsur, D. J. Saltman, and theory, J. W. Gray, Editor G. B. Seligman, Editors 31 Axiomatic set theory, James E. 14 Lectures on Nielsen fixed point theory, Baumgartner, Donald A. Martin, and Boju Jiang Saharan Shelah, Editors 15 Advanced analytic number theory. 32 Proceedings of the conference on Part 1: Ramification theoretic methods, Banach algebras and several complex Carlos J. Moreno variables, F. Greenleaf and D. Gulick, 16 Complex representations of GL(2, K) for Editors finite fields K, llya Piatetski-Shapiro 33 Contributions to group theory, 17 Nonlinear partial differential equations, Kenneth I. Appel, John G. Ratcliffe, and Joel A. Smaller, Editor Paul E. Schupp, Editors 18 Fixed points and nonexpansive 34 Combinatoric& and algebra, Curtis mappings, Robert C. Sine, Editor Greene, Editor Titles in This Series

Volume 35 Four-manifold theory, Cameron Gordon 53 The Selberg trace formula and related and Robion Kirby, Editors topics, Dennis A. Hejhal, Peter Sarnak, 36 Group actions on manifolds, Reinhard and Audrey Anne Terras, Editors Schultz, Editor 54 Differential analysis and infinite 37 Conference on algebraic topology in dimensional spaces, Kondagunta honor of Peter Hilton, Renzo Piccinini Sundaresan and Srinivasa Swaminathan, and Denis Sjerve, Editors Editors 38 Topics in complex analysis, Dorothy 55 Applications of algebraic K-theory to Browne Shaffer, Editor algebraic geometry and number theory, 39 Errett Bishop: Reflections on him and Spencer J. Bloch, R. Keith Dennis, Eric his research, Murray Rosenblatt, Editor M. Friedlander, and Michael R. Stein, 40 Integral bases for affine Lie algebras Editors and their universal enveloping algebras, 56 Multiparameter bifurcation theory, David Mitzman Martin Golubitsky and 41 Particle systems, random media and John Guckenheimer, Editors large deviations, Richard Durrett, Editor 57 Combinatorics and ordered sets, 42 Classical real analysis, Daniel Ivan Rival, Editor Waterman, Editor 58.1 The Lefschetz centennial conference. 43 Group actions on rings, Susan Proceedings on algebraic geometry, Montgomery, Editor D. Sundararaman, Editor 44 Combinatorial methods in topology and 58.11 The Lefschetz centennial conference. algebraic geometry, John R. Harper Proceedings on algebraic topology, and Richard Mandelbaum, Editors S. Gitler, Editor 45 Finite groups-coming of age, John 58.111 The Lefschetz centennial conference. McKay, Editor Proceedings on differential equations, 46 Structure of the standard modules A. Verjovsky, Editor for the affine AP), 59 · Function estimates, J. S. Marron, Editor James Lepowsky and Mirko Prime Nonstrictly hyperbolic conservation 47 Linear algebra and its role in 60 laws, Barbara Lee Keyfitz and systems theory, Richard A. Brualdi, Herbert C. Kranzer, Editors David H. Carlson, Biswa Nath Datta, Charles R. Johnson, and Robert J. 61 Residues and traces of differential Plemmons, Editors forms via Hochschild homology, 48 Analytic functions of one complex Joseph Lipman variable, Chung-chun Yang and 62 Operator algebras and mathematical Chi-tai Chuang, Editors physics, Palle E. T. Jorgensen and 49 Complex differential geometry and PaulS. Muhly, Editors nonlinear differential equations, 63 Integral geometry, Robert L. Bryant, Yum-Tong Siu, Editor Victor Guillemin, Sigurdur Helgason, and 50 Random matrices and their R. 0. Wells, Jr., Editors applications, Joel E. Cohen, 64 The legacy of Sonya Kovalevskaya, Harry Kesten, and Charles M. Newman, , Editor Editors 65 Logic and combinatorics, Stephen G. 51 Nonlinear problems in geometry, Simpson, Editor Dennis M. DeTurck, Editor 52 Geometry of normed linear spaces, 66 Free group rings, Narian Gupta R. G. Bartle, N. T. Peck, A. L. Peressini, 67 Current trends in arithmetical algebraic and J. J. Uhl, Editors geometry, Kenneth A. Ribet, Editor Titles in This Series

Volume 68 Differential geometry: The interface 86 Representation theory and number between pure and applied mathematics, theory in connection with the local Mladen Luksic, Clyde Martin, and Langlands conjecture, J. Ritter, Editor William Shadwick, Editors 87 Abelian group theory, Laszlo Fuchs, 69 Methods and applications of Rudiger Gobel, and Phillip Schultz, mathematical logic, Walter A. Carnielli Editors and Luiz Paulo de Alcantara, Editors 88 Invariant theory, R. Fossum, 70 Index theory of elliptic operators, W. Haboush, M. Hochster, and foliations, and operator algebras, V. Lakshmibai, Editors Jerome Kaminker, Kenneth C. Millett, 89 Graphs and algorithms, R. Bruce and Claude Schochet, Editors Richter, Editor 71 Mathematics and general relativity, 90 Singularities, Richard Randell, Editor James A. Isenberg, Editor 91 Commutative harmonic analysis, 72 Fixed point theory and its applications, David Colella, Editor R. F. Brown, Editor 92 Categories in computer science and 73 Geometry of random motion, Rick logic, John W. Gray and Andre Scedrov, Durrett and Mark A. Pinsky, Editors Editors 93 Representation theory, group rings, and 74 Geometry of group representations, coding theory, M. Isaacs, A. Lichtman, William M. Goldman and Andy R. Magid, D. Passman, S. Sehgal, N.J. A. Sloane, Editors and H. Zassenhaus, Editors 75 The finite calculus associated with 94 Measure and measurable dynamics, Bessel functions, Frank M. Cholewinski R. Daniel Mauldin, R. M. Shortt, and 76 The structure of finite algebras, Cesar E. Silva, Editors David C. Hobby and Ralph Mckenzie 95 Infinite algebraic extensions of finite 77 Number theory and its applications in fields, Joel V. Brawley and George E. China, Wang Yuan, Yang Chung-chun, Schnibben and Pan Chengbiao, Editors 96 Algebraic topology, Mark Mahowald 78 Braids, Joan S. Birman and Anatoly and Stewart Priddy, Editors Libgober, Editors 97 Dynamics and control of multibody 79 Regular differential forms, Ernst Kunz systems, J. E. Marsden, P. S. and Rolf Waldi Krishnaprasad, and J. C. Simo, Editors 80 Statistical inference from stochastic 98 Every planar map is four colorable, processes, N. U. Prabhu, Editor Kenneth Appel and Wolfgang Haken 99 The connection between infinite 81 Hamiltonian dynamical systems, dimensional and finite dimensional Kenneth R. Meyer and Donald G. Saari, dynamical systems, Basil Nicolaenko, Editors Ciprian Foias, and Roger Temam, 82 Classical groups and related topics, Editors Alexander J. Hahn, Donald G. James, 100 Current progress in hyperbolic systems: and Zhe-xian Wan, Editors Riemann problems and computations, 83 Algebraic K-theory and algebraic W. Brent Lindquist, Editor number theory, Michael R. Stein and 101 Recent developments in geometry, R. Keith Dennis, Editors S.-Y. Cheng, H. Choi, and Robert E. 84 Partition problems in topology, Greene, Editors Stevo Todorcevic 102 Primes associated to an ideal, 85 Banach space theory, Bor-Luh Lin, Stephen McAdam Editor 103 Coloring theories, Steve Fisk Titles in This Series

Volume 104 Accessible categories: The foundations of categorical model theory, Michael Makkai and Robert Pare 105 Geometric and topological invariants of elliptic operators, Jerome Kaminker, Editor 106 Logic and computation, Wilfried Sieg, Editor 107 Harmonic analysis and partial differential equations, Mario Milman and Tomas Schonbek, Editors 108 Mathematics of nonlinear science, Melvyn S. Berger, Editor 109 Combinatorial group theory, Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors 110 Lie algebras and related topics, and J. Marshall Osborn, Editors 111 Finite geometries and combinatorial designs, Earl S. Kramer and Spyros S. Magliveras, Editors 112 Statistical analysis of measurement error models and applications, Philip J. Brown and Wayne A. Fuller, Editors 113 Integral geometry and tomography, Eric Grinberg and Eric Todd Quinto, Editors 114 Mathematical developments arising from linear programming, Jeffrey C. Lagarias and Michael J. Todd, Editors 115 Statistical multiple integration, Nancy Flournoy and Robert K. Tsutakawa, Editors 116 Algebraic geometry: Sundance 1988, Brian Harbourne and Robert Speiser, Editors Algebraic Geometry: Sundance 1988 CoNTEMPORARY MATHEMATICS

116

Algebraic Geometry: Sundance 1988

Proceedings of a Conference on Algebraic Geometry held July 18-23, 1988 with support from Brigham Young University and the National Science Foundation

Brian Harbourne Robert Speiser Editors

American Mathematical Society Providence, Rhode Island EDITORIAL BOARD

Richard W. Beals, managing editor Sylvain E. Cappel! Craig Huneke Michael E. Taylor

The Conference on Algebraic Geometry was held at Sundance, Utah on July 18-23, 1988 with support from Brigham Young University and the National Science Foundation, Grant DMS-8719233.

1980 Mathematics Subject Classification (1985 Revision). Primary 14C05, 14020, 14H10, 14C20, 14C22, 14J26, 14E30, 14B05, 14N05, 14C17, 14B07, 14N10, 14J25, 14E25, 14E17.

Library of Congress Cataloging-in-Publication Data The Conference on Algebraic Geometry (1988: Sundance, Utah) Algebraic geometry: Sundance 1988: proceedings of a Conference on Algebraic Geometry held July 18-23, 1988/Brian Harbourne and Robert Speiser, editors. p. cm.-(Contemporary mathematics; 116) ISBN 0-8218-5124-1 (alk. paper) 1. Geometry, Algebraic-Congresses. I. Harbourne, Brian, 1955-. II. Speiser, R. (Robert) Ill. Title. IV. Series: Contemporary mathematics (American Mathematical Society); v. 116. QA564.C657 1988 90-29884 516.3'5-dc20 CIP

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Preface Xl Hurwitz Surfaces with Nontrivial Divisors STEVEN DIAZ AND RoN DoNAGI Normal Sheaves of Linear Systems on Curves LAWRENCE EIN 9 An Overview of the Geometry of Algebraic Fermi Curves D. GIESEKER, H. KNORRER, AND E. TRUBOWITZ 19 Automorphisms of Cuspidal K3-Like Surfaces BRIAN HARBOURNE 47 Small Resolutions of Gorenstein Threefold Singularities SHELDON KATZ 61 Multiple Tangents of Smooth Plane Curves (after Kaji) STEVEN L. KLEIMAN 71 Enumerative Geometry of Nonsingular Plane Cubics STEVEN L. KLEIMAN AND ROBERT SPEISER 85 The Gaussian Map for Certain Planar Graph Curves RICK MIRANDA 115 Absence of the Veronese from Smooth Threefolds in 1? 5 Ziv RAN 125 Limits of Conormal Schemes ROBERT SPEISER 129

ix Preface

This volume represents the proceedings of the 1988 Sundance conference on algebraic curves and related varieties, July 18-23, 1988. Some papers here represent actual conference lectures, or report research carried out at Sundance, while others describe related work done elsewhere.

The beauty of the mountains, the fine weather, the particularly comfortable working environment, and the participants' energy combined to make this conference especially memorable. It is a pleasure to thank everyone who took part and express gratitude as well to Brigham Young University and the National Science Foundation (grant DMS-8719233) for strong support.

Brian Harbourne Robert Speiser

xi .-.

ISBN 0-8218-5124-1