CONTEMPORARY MATHEMATICS 123

Enumerative Algebraic Geometry Proceedings of tne 1989 Zeuthen Symposium http://dx.doi.org/10.1090/conm/123

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Volume 1 Markov random fields and their 13 Algebraists' homage: Papers in applications, Ross Kindermann and ring theory and related topics, J. Laurie Snell S. A. Amitsur, D. J. Saltman, and 2 Proceedings of the conference on G. B. Seligman, Editors integration, topology, and geometry in 14 Lectures on Nielsen fixed point theory, linear spaces, William H. Graves, Editor Boju Jiang 3 The closed graph and P-closed 15 Advanced analytic number theory. graph properties in general topology, Part 1: Ramification theoretic methods, T. R. Hamlett and L. L. Herrington Carlos J. Moreno 4 Problems of elastic stability and 16 Complex representations of GL(2, K) for vibrations, Vadim Komkov, Editor finite fields K, llya Piatetski-Shapiro 5 Rational constructions of modules 17 Nonlinear partial differential equations, for simple Lie algebras, George B. Joel A. Smaller, Editor Seligman 18 Fixed points and nonexpansive 6 Umbral calculus and Hopf algebras, mappings, Robert C. Sine, Editor Robert Morris, Editor 19 Proceedings of the Northwestern 7 Complex contour integral homotopy theory conference, Haynes representation of cardinal spline R. Miller and Stewart B. Priddy, Editors functions, Walter Schempp 20 Low dimensional topology, Samuel J. 8 Ordered fields and real algebraic Lomonaco, Jr., Editor geometry, D. W. Dubois and T. Recio, Editors 21 Topological methods in nonlinear functional analysis, S. P. Singh, 9 Papers in algebra, analysis and S. Thomaier, and B. Watson, Editors statistics, R. Lidl, Editor 22 Factorizations of b" ± 1, b = 2, 10 Operator algebras and K-theory, 3, 5, 6, 7, 10, 11,12 up to high Ronald G. Douglas and Claude powers, John Brillhart, D. H. Lehmer, Schochet, Editors J. L. Selfridge, Bryant Tuckerman, and 11 Plane ellipticity and related problems, S. S. Wagstaff, Jr. Robert P. Gilbert, Editor 23 Chapter 9 of Ramanujan's second 12 Sympocium on algebraic topology in notebook-Infinite series identities, honor of Jose Adem, Samuel Gitler, transformations, and evaluations, Editor Bruce C. Berndt and Padmini T. Joshi Titles in This Series

Volume 24 Central extensions, Galois groups, and 43 Group actions on rings, Susan ideal class groups of number fields, Montgomery, Editor A. Frohlich 44 Combinatorial methods in topology and 25 Value distribution theory and its algebraic geometry, John R. Harper applications, Chung-Chun Yang, Editor and Richard Mandelbaum, Editors 26 Conference in modern analysis 45 Finite groups-coming of age, John and probability, Richard Beals, McKay, Editor Anatole Beck, Alexandra Bellow, and Arshag Hajian, Editors 46 Structure of the standard modules for the affine Lie algebra A~ 1 ), 27 Microlocal analysis, M. Salah Baouendi, James Lepowsky and Mirko Prime Richard Beals, and Linda Preiss Rothschild, Editors 47 Linear algebra and its role in systems theory, Richard A. Brualdi, 28 Fluids and plasmas: geometry and David H. Carlson, Biswa Nath Datta, dynamics, Jerrold E. Marsden, Editor Charles R. Johnson, and Robert J. 29 Automated theorem proving, W. W. Plemmons, Editors Bledsoe and Donald Loveland, Editors 48 Analytic functions of one complex 30 Mathematical applications of category variable, Chung-chun Yang and theory, J. W. Gray, Editor Chi-tai Chuang, Editors 31 Axiomatic set theory, James E. Baumgartner, Donald A. Martin, and 49 Complex differential geometry and Saharan Shelah, Editors nonlinear differential equations, Yum-Tong Siu, Editor 32 Proceedings of the conference on Banach algebras and several complex 50 Random matrices and their variables, F. Greenleaf and D. Gulick, applications, Joel E. Cohen, Editors Harry Kesten, and Charles M. Newman, Editors 33 Contributions to group theory, Kenneth I. Appel, John G. Ratcliffe, and 51 Nonlinear problems in geometry, Paul E. Schupp, Editors Dennis M. DeTurck, Editor 34 Combinatoric& and algebra, Curtis 52 Geometry of normed linear spaces, Greene, Editor R. G. Bartle, N. T. Peck, A. L. Peressini, 35 Four-manifold theory, Cameron Gordon and J. J. Uhl, Editors and Robion Kirby, Editors 53 The Selberg trace formula and related 36 Group actions on manifolds, Reinhard topics, Dennis A. Hejhal, Peter Sarnak, Schultz, Editor and Audrey Anne Terras, Editors 37 Conf.erence on algebraic topology in 54 Differential analysis and infinite honor of Peter Hilton, Renzo Piccinini dimensional spaces, Kondagunta and Denis Sjerve, Editors Sundaresan and Srinivasa Swaminathan, 38 Topics in complex analysis, Dorothy Editors Browne Shaffer, Editor 55 Applications of algebraic K-theory to 39 Errett Bishop: Reflections on him and algebraic geometry and number theory, his research, Murray Rosenblatt, Editor Spencer J. Bloch, R. Keith Dennis, Eric 40 Integral bases for affine Lie algebras M. Friedlander, and Michael R. Stein, and their universal enveloping algebras, Editors David Mitzman 56 Multiparameter bifurcation theory, 41 Particle systems, random media and Martin Golubitsky and large deviations, Richard Durrett, Editor John Guckenheimer, Editors 42 Classical real analysis, Daniel 57 Combinatorics and ordered sets, Waterman, Editor Ivan Rival, Editor Titles in This Series

Volume 58.1 The Lefschetz centennial conference. 74 Geometry of group representations, Proceedings on algebraic geometry, William M. Goldman and Andy R. Magid, D. Sundararaman, Editor Editors 58.11 The Lefschetz centennial conference. 75 The finite calculus associated with Proceedings on algebraic topology, Bessel functions, Frank M. Cholewinski S. Gitler, Editor 76 The structure of finite algebras, 58.111 The Lefschetz centennial conference. David C. Hobby and Ralph Mckenzie Proceedings on differential equations, 77 Number theory and its applications in A. Verjovsky, Editor China, Wang Yuan, Yang Chung-chun, 59 Function estimates, J. S. Marron, Editor and Pan Chengbiao, Editors 78 Braids, Joan S. Birman and Anatoly 60 Nonstrictly hyperbolic conservation Libgober, Editors laws, Barbara Lee Keyfitz and Herbert C. Kranzer, Editors 79 Regular differential forms, Ernst Kunz and Rolf Waldi 61 Residues and traces of differential 80 Statistical inference from stochastic forms via Hochschild homology, processes, N. U. Prabhu, Editor Joseph Lipman 81 Hamiltonian dynamical systems, 62 Operator algebras and mathematical Kenneth R. Meyer and Donald G. Saari, physics, Palle E. T. Jorgensen and Editors Paul S. Muhly, Editors 82 Classical groups and related topics, 63 Integral geometry, Robert L. Bryant, Alexander J. Hahn, Donald G. James, Victor Guillemin, Sigurdur Helgason, and and Zhe-xian Wan, Editors R. 0. Wells, Jr., Editors 83 Algebraic K-theory and algebraic 64 The legacy of Sonya Kovalevskaya, number theory, Michael R. Stein and Linda Keen, Editor R. Keith Dennis, Editors 65 Logic and combinatorics, Stephen G. 84 Partition problems in topology, Simpson, Editor Stevo Todorcevic 66 Free group rings, Narian Gupta 85 Banach space theory, Bor-Luh Lin, Editor 67 Current trends in arithmetical algebraic geometry, Kenneth A. Ribet, Editor 86 Representation theory and number theory in connection with the local 68 Differential geometry: The interface Langlands conjecture, J. Ritter, Editor between pure and applied mathematics, 87 Abelian group theory, Laszlo Fuchs, Mladen Luksic, Clyde Martin, and Rudiger Gobel, and Phillip Schultz, William Shadwick, Editors Editors 69 Methods and applications of 88 Invariant theory, R. Fossum, mathematical logic, Walter A. Carnielli W. Haboush, M. Hochster, and and Luiz Paulo de Alcantara, Editors V. Lakshmibai, Editors 70 Index theory of elliptic operators, 89 Graphs and algorithms, R. Bruce foliations, and operator algebras, Richter, Editor Jerome Kaminker, Kenneth C. Millett, 90 Singularities, Richard Randell, Editor and Claude Schochet, Editors 91 Commutative harmonic analysis, 71 Mathematics and general relativity, David Colella, Editor James A. Isenberg, Editor 92 Categories in computer science and 72 Fixed point theory and its applications, logic, John W. Gray and Andre Scedrov, R. F. Brown, Editor Editors 73 Geometry of random motion, Rick 93 Representation theory, group rings, and Durrett and Mark A. Pinsky, Editors coding theory, M. Isaacs, A. Lichtman, Titles in This Series

Volume D. Passman, S. Sehgal, N.J. A. Sloane, 111 Finite geometries and combinatorial and H. Zassenhaus, Editors designs, Earl S. Kramer and Spyros S. 94 Measure and measurable dynamics, Magliveras, Editors R. Daniel Mauldin, R. M. Shortt, and 112 Statistical analysis of measurement Cesar E. Silva, Editors error models and applications, 95 Infinite algebraic extensions of finite Philip J. Brown and Wayne A. Fuller, fields, Joel V. Brawley and George E. Editors Schnibben 1131ntegral geometry and tomography, 96 Algebraic topology, Mark Mahowald Eric Grinberg and Eric Todd Quinto, and Stewart Priddy, Editors Editors 97 Dynamics and control of multibody 114 Mathematical developments arising systems, J. E. Marsden, P. S. from linear programming, Jeffrey C. Krishnaprasad, and J. C. Simo, Editors Lagarias and Michael J. Todd, Editors 98 Every planar map is four colorable, 115 Statistical multiple integration, Kenneth Appel and Wolfgang Haken Nancy Flournoy and Robert K. Tsutakawa, Editors 99 The connection between infinite 116 Algebraic geometry: Sundance 1988, dimensional and finite dimensional Brian Harbourne and Robert Speiser, dynamical systems, Basil Nicolaenko, Editors Ciprian Foias, and Roger Temam, Editors 117 Continuum theory and dynamical systems, Morton Brown, Editor 100 Current progress in hyperbolic systems: Riemann problems and computations, 118 Probability theory and its applications in W. Brent Lindquist, Editor China, Van Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors 101 Recent developments in geometry, S.-Y. Cheng, H. Choi, and Robert E. 119 Vision geometry, Robert A. Melter, Azriel Rosenfeld, and Prabir Greene, Editors Bhattacharya, Editors 102 Primes associated to an ideal, 120 Selfadjoint and nonselfadjoint operator Stephen McAdam algebras and operator theory, 103 Coloring theories, Steve Fisk Robert S. Doran, Editor 104 Accessible categories: The foundations 121 Spinor construction of vertex operator of categorical model theory, Michael algebras, triality, and E~ 1 ), Alex J. Makkai and Robert Pare Feingold, Igor B. Frenkel, and John F. X. 105 Geometric and topological invariants Ries of elliptic operators, Jerome Kaminker, 122 Inverse scattering and applications, Editor D. H. Sattinger, C. A. Tracy, and S. 106 Logic and computation, Wilfried Sieg, Venakides, Editors Editor 123 Enumerative algebraic geometry, 107 Harmonic analysis and partial Steven L. Kleiman and Anders Thorup, differential equations, Mario Milman Editors 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 11 0 Lie algebras and related topics, Georgia Benkart and J. Marshall Osborn, Editors Enumerative Algebraic Geometry Proceedings of the 1989 Zeuthen Symposium

CoNTEMPORARY MATHEMATICS

123

Enumerative Algebraic Geometry Proceedings of the 1989 Zeuthen Symposium

Proceedings of a Symposium held July 30-August 6, 1989, with support from the Danish Natural Science Research Council, Augustinus Fonden, Niels Bohr Legatet, Carlsbergfondet, and Unimerco's Fond.

Steven L. Kleiman Anders Thorup Editors

American Mathematical Society Providence, Rhode Island EDITORIAL BOARD

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

The Zeuthen Symposium was held at the Mathematical Institute of the Univer- sity of Copenhagen, Copenhagen, Denmark, on July 30-August 6, 1989 with sup- port from the Danish Natural Science Research Council, Grant 11-7428, Augustinus Fonden, Niels Bohr Legatet, Carlsbergfondet, and Unimerco's Fond.

1991 Mathematics Subject Classification. Primary 14-06, 14N10, 14N05, 14C17.

Library of Congress Cataloging-in-Publication Data Zeuthen Symposium (1989: Mathematical Institute of the University of Copenhagen) Enumerative algebraic geometry: proceedings of the 1989 Zeuthen Symposium 1 Steven L. Kleiman and Anders Thorup, editors. p. cm.-(Contemporary mathematics; 123) Conference held to celebrate the 150th anniversary of the birth of the Danish mathematician. Includes bibliograhical references. ISBN 0-8218-5131-4 1. Geometry, Algebraic-Congresses. 2. Combinatorial enumeration problems-Congresses. 3. Intersection theory-Congresses. I. Zeuthen, H. G. (Hieronymus Georg), 1839-1920. II. Kleiman, Steven L. Ill. Thorup, Anders, 1943-. IV. Title. V. Series: Contemporary mathematics (American Mathematical Society); v. 123. QA564.Z48 1989 91-37558 516.315-dc20 CIP

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Preface xi

Hieronymus Georg Zeuthen (1839-1920) STEVEN L. KLEIMAN 1

How many smooth plane cubics with given j-invariant are tangent to 8 lines in general position? PAOLO ALUFFI 15

Triple and quadruple contact of plane curves SusAN JANE COLLEY AND GARY KENNEDY 31

Secant lines of smooth projective curves: an infinitesimal study of the symmetric products MARC COPPENS AND TRYGVE JOHNSEN 61

New bounds for Fermat curves over finite fields ABRAMO HEFEZ AND NEUZA KAKUTA 89

Discriminants and limits of duals of plane curves SHELDON KATZ 99 On the inseparability of the Gauss map STEVEN KLEIMAN AND RAGNI PlENE 107 The Brill-Segre formula for families of curves DAN LAKSOV AND ANDERS THORUP 131 Duality for elliptic normal surface scrolls RAQUEL MALLAVIBARRENA AND RAGNI PlENE 149 A note on the Riemann-Kempf theorem for curves ARTHUR MATTUCK 161

On the geometry of nodal plane cubics: the condition p J. M. MIRET AND s. XAMBO DESCAMPS 169

ix X CONTENTS

Characteristic numbers of degeneracy loci ADAM PARUSINSKI AND PIOTR PRAGACZ 189 On Halphen's first formula C. PROCESI AND S. XAMBO 0ESCAMPS 199 Negative intersection multiplicities on singular varieties PAUL ROBERTS 213 Triple contact formulas in IP'3 FRANcEse RossELL6 223 Elliptic quartic curves in a generic quintic threefold ISRAEL V AINSENCHER 247 Characteristic numbers of plane curves: An excess intersection theoretical approach LEENDERT J. VAN GASTEL 259 Degenerations of conormal varieties LEENDERT J. VANGASTEL 267 Preface

The year 1989 was the 150th anniversary of the birth of the great Danish math- ematician Hieronymous Georg Zeuthen. Although he spent nearly all of his long professional life in Copenhagen, Zeuthen's name is known to every algebraic geometer throughout the world because of his discovery of a basic invariant of surfaces. However, Zeuthen did a lot of other fundamental research in inter- section theory, enumerative geometry, and the projective geometry of curves and surfaces. Zeuthen's extraordinary devotion to his subject, his characteristic depth, thoroughness, and clarity of thought, and his precise and succinct writing are truely inspiring. During the last ten years or so, algebraic geometers have re-examined Zeuthen's work, derived inspiration from it, and carried it further. The 1989 Zeuthen Symposium presented a historic opportunity for mathemati- cians to gather and examine those areas in contemporary mathematical research which have evolved from the fruitful ideas and tendencies of Zeuthen's original work. The Zeuthen Symposium was held from July 30 to August 6, 1989 at the Mathematical Institute of the University of Copenhagen. These proceedings contain papers presented at the symposium, and papers inspired by it; all these papers were refereed, and are in final form. The Zeuthen Symposium received financial support from the Danish Natural Science Research Council under grant 11-7428, and from the Augustinus Fonden, the Niels Bohr Legatet, the Carlsbergfondet, and Unimerco's Fond. On behalf of the symposium, the organizers- Hans-Bj0rn Foxby, Steven L. Kleiman, and Anders Thorup - would like to express their gratitude to the referees and the sponsors.

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