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Computational Algebraic and Analytic Geometry 572 Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors American Mathematical Society Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors 572 Computational Algebraic and Analytic Geometry AMS Special Sessions on Computational Algebraic and Analytic Geometry for Low-Dimensional Varieties January 8, 2007: New Orleans, LA January 6, 2009: Washington, DC January 6, 2011: New Orleans, LA Mika Seppälä Emil Volcheck Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2010 Mathematics Subject Classification. Primary 14HXX, 30FXX, and 68WXX. Library of Congress Cataloging-in-Publication Data Computational algebraic and analytic geometry : AMS special sessions on computational algebraic and analytic geometry for low-dimensional varieties, January 8, 2007, New Orleans, LA, January 6, 2009, Washington, DC, [and] January 6, 2011, New Orleans, LA / Mika Sepp¨al¨a, Emil Volcheck, editors. p. cm. — (Contemporary mathematics ; v. 572) Includes bibliographical references. ISBN 978-0-8218-6869-0 (alk. paper) 1. Curves, Algebraic–Data processing–Congresses. 2. Riemann surfaces–Congresses. I. Sepp¨al¨a, Mika. II. Volcheck, Emil, 1966– QA565.C658 2012 512’.5–dc23 2012009188 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 Contents Preface vii Large Hyperbolic Polygons and Hyperelliptic Riemann Surfaces Anthony Arnold and Klaus-Dieter Semmler 1 On Isolated Strata of Pentagonal Riemann Surfaces in the Branch Locus of Moduli Spaces Gabriel Bartolini, Antonio F. Costa, and Milagros Izquierdo 19 Finite Group Actions of Large Order on Compact Bordered Surfaces E. Bujalance, F. J. Cirre, and M. D. E. Conder 25 Surfaces of Low Degree Containing a Canonical Curve Izzet Coskun 57 Ideals of Curves Given by Points E. Fortuna, P. Gianni, and B. Trager 71 Non-genera of Curves with Automorphisms in Characteristic p Darren Glass 89 Numerical Schottky Uniformizations of Certain Cyclic L-gonal Curves Ruben´ A. Hidalgo and Mika Seppal¨ a¨ 97 Generalized Lantern Relations and Planar Line Arrangements Eriko Hironaka 113 Effective p-adic Cohomology for Cyclic Cubic Threefolds Kiran S. Kedlaya 127 Generating Sets of Affine Groups of Low Genus K. Magaard, S. Shpectorov, and G. Wang 173 Classification of Algebraic ODEs with Respect to Rational Solvability L. X. Chauˆ Ngo,ˆ J. Rafael Sendra, and Franz Winkler 193 Circle Packings on Conformal and Affine Tori Christopher T. Sass, Kenneth Stephenson, and G. Brock Williams 211 Effective Radical Parametrization of Trigonal Curves Josef Schicho and David Sevilla 221 v Preface Distinct communities of mathematicians have grown around analytical and algebraic approaches to geometry. Even though both approaches are deeply con- nected through results such as Chow’s Theorem and GAGA, mathematicians in- frequently collaborate across these communities. Computational methods make these connections explicit and increase our understanding of geometry in ways not possible when each approach is pursued as its own form of pure mathematics. In this way, computational methods help bring together these different communities of mathematicians. Uniformization of Riemann surfaces is a prime example of a topic where com- putational methods are bringing important new insights. In the late 19th and early 20th centuries, mathematicians such as Burnside, Koebe, Myrberg, Rankin, and Whittaker labored to make uniformization explicit, developing numerical tech- niques even when no computers were available. Their work is considered to be a crowning achievement of geometry in that era. The Abel-Jacobi and Torelli Theo- rems represent another prominent example of a theory that relates analytical and algebraic representations of geometric objects, in this case, relating complex lattices to curves and their Jacobians. During the last twenty years, practical numerical and symbolic computations have become commonplace and possible for anybody. This has given new life to some of the old ideas, and has led to new approaches to some of the classical problems. Here are some examples of such work. Uniformization has advanced in both theory and practice through the develop- ment of effective computational methods for special cases of the problem. Ideally one would like to find explicit symbolic methods to pass, for example, from an algebraic plane curve, given by a polynomial, to a uniformization of the curve in question. In the case of genus one, the symbolic approach is part of the classical analysis of elliptic curves. For curves of higher genus, symbolic methods have suc- ceeded in special cases only. Numeric methods have yielded more general results, but a solution to the general case still looms far in the future. These numeric methods lead one to study, for example, algebraic curves given by approximations of the actual polynomials defining the curve, which is a major topic in numerical algebraic geometry. An example of a completely new theory whose development was supported by computational methods is a discrete version of the Riemann mapping theo- rem offered by circle packings. Research on explicit methods connecting Riemann surfaces and their corresponding Fuchsian groups has also benefited from compu- tational methods. This series is also inspired by work of Curtis McMullen, who, in vii viii PREFACE his AMS Colloquium Lectures in 2000, connected dynamics on a Riemann surface to rational points on the corresponding algebraic curve. This volume is a collection of research papers on computational methods in algebraic and analytic geometry. It has its roots in the series of AMS Special Sessions on Computational Algebraic and Analytic Geometry that have taken place at the Joint AMS-MAA National Meetings every odd year since 1999, and in the large European research projects that coordinated the work in this area during 1991–1996. Usually AMS Special Sessions and other similar meetings are characterized by the methods used in the papers presented. This volume and the preceding AMS Special Sessions form an exception to this rule: papers published here and those presented earlier in the Special Sessions entitled Computational Algebraic and Analytic Geometry on Low-Dimensional Varieties have, as their unifying factor, the same object of study. Compact Riemann surfaces are algebraic curves. They are also characterized by their Jacobian variety and their Fuchsian group. Hence the same object can be studied by a variety of methods: analytic, algebraic, and geometric. It is this extraordinary variety of methods that makes this area challenging, interesting, and very fertile. The editors are grateful for the contributions of the authors, and the referees who helped to create this volume. The editors thank all the sponsoring institutions that helped to advance research in this field. Most importantly, the editors thank the American Mathematical Society and its expert publishing officers. Mika Sepp¨al¨a and Emil Volcheck Contemporary Mathematics Volume 572, 2012 http://dx.doi.org/10.1090/conm/572/11364 Large hyperbolic polygons and hyperelliptic Riemann surfaces Anthony Arnold and Klaus–Dieter Semmler Abstract. The following pages will present large hyperbolic polygons as a use- ful tool to study hyperelliptic Riemann surfaces. A large hyperbolic polygon is a set of vertices in the hyperbolic plane, where one can draw lines, one through each vertex, such that for any given such line all others lie entirely in the same half-space with respect to the given one. Identifying a vertex with the half-turn around it, the vertices
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