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Applications of Algebraic Geometry to Coding Theory, Physics and Computation

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I. Life and Behavioural Sciences II. Mathematics, Physics and Chemistry III. Computer and Systems Science IV. Earth and Environmental Sciences

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Series II: Mathematics, Physics and Chemistry - Vol. 36

Applications of Aigebraic Geometry to Coding Theory, Physics and Computation

edited by

Ciro Ciliberto Oipartimenlo di Matematica, Universita di Roma "Tor Vergata", Roma, naly

Friedrich Hirzebruch Max Planck Institut für Mathematik, Bonn, Germany

Rick Miranda Deparlment 01 Malhemalics. Colorado Slale University. Fort Collins, CO, U.S.A.

and

Mina Teicher Department 01 Malhemal ics and Computer Seienee, Bar-lIan University, Ramal Gan, Israel

Springer-Science+Business Med ia, B_V_

Praceedings of the NATO Advanced Research Workshop on Applications of Aigebraic Geornetry to Coding Theory, Physics and Computation Eilat, Israel 25th February to 1 March 2001

A C.I.P. Catalogue record for this book is available fram the Library of Congress.

ISBN 978-1-4020-0005-8 ISBN 978-94-010-1011-5 (eBook) DOI 10.1007/978-94-010-1011-5

Printed on acid-free paper

A" Rights Reserved © 2001 Springer Science+Business Media Dordrecht Origina"y published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1st edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission fram the copyright owner.

CONTENTS

Preface

List of participants

List of contributors

I. Burban, Yu. Drozd and G.-M. Greuel: Vector bundles on singular projective curves

A. Calabri: On double planes with Kodaira dimension zero

L. Chiantini, F. Cioffi and F. Orecchia: Computing minimal generators of ideals of elliptic curves

C. Ciliberto and R. Miranda: The Segre and Harbourne-Hirschowitz conjectures

C. Ciliberto, R. Miranda and M. Teicher:

VII

IX

XIII

17

23

37

Pillow degenerations ofK3 swfaces 53

W. Decker and F.-O. Schreyer: Computational algebraic geometry today 65

M. Fryers, J.Y. Kaminski and M. Teicher: Some applications of algebraic curves to computational vision 121

G. van der Geer: Coding theory and algebraic curves over finite fields

G.-M. Greuel, C. Lossen and M. Schulze: Three algorithms in algebraic geometry, coding theory and singularity theOlY

K. Hulek and J. Spandaw: Counting points on Calabi-Yau threefolds

E.Izadi: Subvarieties of abelian varieties

A. Libgober: Characteristic varieties of algebraic curves

139

161

195

207

215

VI CONTENTS

R. Livne: Communication networks and Hilbert modular forms 255

Th. Peternel1: Compact Kahler threefhlds with small Picard numbers 271

R. Schoof: Abelian varieties over thefield of the 20th roots of unity that have good reduction everywhere 291

AJ. Sommese, J. Verschelde and C.w. Wampler: Using monodromy to decompose solution sets of polynomial systems into irreducihle components 297

B. Szendroi: D~ffeom01phisms andfamilies of Fourier-Mukai transforms in mirror .symmetlY 317

PREFACE

A NATO Advanced Research Workshop on Applications of Algebraic Geom­etl)! to Coding TheOl)!, Physics and Computation, was held at the Eilat Princess Hotel in Eilat, Israel, between February 25 and March 1,2001. The purpose of the Workshop was to bring together researchers interested in significant applications of algebraic geometry, as well as in fundamental research, for a week of presen­tations of recent contributions and proposed new problems, as well as intensive discussions of possible collaborative efforts.

Application areas discussed explicitly during lectures of the workshop in­cluded computational algebra, singularity theory algorithms, unified field theory, numerical solutions of polynomial systems, coding theory, communication net­works, and computer vision. A demonstration of the computer algebra system SINGULAR was kindly presented by Prof. G.-M. Greuel, who is one of its main developers.

Lectures on more fundamental aspects of algebraic geometry included expo­sitions related to counting points on varieties over finite fields, Mori theory, real and pseudo-holomorphic curves, linear systems, curves on abelian varieties and abelian varieties with everywhere good reduction, elliptic genera, fundamental groups of complements of cuspidal curves, configuration spaces, and bitangent reconstruction of plane curves. Many of the lecturers have submitted written ac­counts of their presentations to this volume, and other related papers by several participants have also been included.

The variety of application areas, and the array of algebro-geometric techniques being utilized more and more frequently, emerged as a main theme of the Work­shop. This was especially brought out at a Round Table Discussion held on the second day, at which many of the participants expressed a perception of renewed enthusiasm for algebraic geometry contributions (both from mathematicians and physicists, engineers, and medical practitioners). In addition, it was recognized that there are still many areas of science and technology which could benefit from an infusion of mathematical ideas and techniques, and that, as working mathe­maticians interested in applications, it is our responsibility to make an effort to

VII

VIII PREFACE

bridge this gap whenever feasible. We believe that the articles in the book reflect this in large measure.

The presence of some interesting surveys makes this volume not simply a collection of papers presented at the Workshop, but also an up-to-date report on the present situation of important research topics in algebraic geometry and its applications.

The organizers and participants especially thank the financial sponsors of the Workshop: the NATO Scientific Affairs Division, the Ministry of Science ofIsrael, the Emmy Noether Research Institute of Mathematics at Bar-Han University and EAGER (the European Union Research Network in Algebraic Geometry).

The editors want to thank drs. A. Calabri, B. KuniavskiI and F. Tovena for their substantial help in the organization of the workshop. Special thanks go to A. Ca­labri for having attended to most of the editing process of the present volume.

The Editors: Ciro Ciliberto Friedrich Hirzebruch Rick Miranda Mina Teicher

LIST OF PARTICIPANTS

NATO Advanced Research Worhhop

Eilaf, hTael, February 25~March 1, 2001

Herbert ABELS abels@mathematik.uni-bielefeld.de University of Bielefeld, Germany

Marco ANDREATTA andreatt@science.unitn.it University of Trento, Italy

Tzachi BEN-ITZHAK benitzi@macs.biu.ac.il Bar-Han University, Israel

Alberto CALABRI calabri@mat.uniroma2.it Univ. of Rome "Tor Vergata", Italy

Maurizio CORNALBA comalba@dimat.unipv.it University of Pavia, Italy

Ron DONAGI donagi@math.upenn.edu University of Pennsylvania, U.S.A.

Francis O. A. GARDEYN fgardeyn@cage.rug.ac.be Ghent University, Belgium

Meirav AMRAM meirav@macs.biu.ac.il Bar-Han University, Israel

Enrico ARBARELLO ea@mat.uniromal.it Univ. of Rome "La Sapienza", Italy

Igor I. BURBAN burban@mathematik.uni-k1.de University of Kaiserslautem, Germany

Ciro CILIBERTO cilibert@mat.uniroma2.it Univ. of Rome "Tor Vergata", Italy

Wolfram DECKER decker@math.uni-sb.de University of Saarlandes, Germany

David GARBER garber@macs.biu.ac.il Bar-Han University, Israel

Gerard van der GEER geer@wins.uva.nl Univ. of Amsterdam, The Netherlands

IX

x LIST OF PARTICIPANTS

Brent GORDON bgordon@nsf.gov National Science Foundation, U.S.A.

Phillip A. GRIFFITHS pg@ias.edu Institute for Advanced Study, U.S.A.

Vasilii A. ISKOVSKIKH iskovsk@mi.ras.ru Steklov Institute, Moscow, Russia

Jeremy KAMINSKI jeremy@cs.huji.ac.il Bar-nan University, Israel

Serguei A. KOUDRIAVTSEV koudr@mi.ras.ru Steklov Institute, Moscow, Russia

Boris KUNIAVSKII kunyav@macs.biu.ac.il Bar-nan University, Israel

Anatoly LIBGOBER libgober@math.uic.edu Univ. oflllinois at Chicago, U.S.A.

RonLIVNE rlivne@math.huji.ac.il The Hebrew Univ., Jerusalem, Israel

Rick MIRANDA miranda@math.colostate.edu Colorado State University, U.S.A.

Claudio PEDRINI pedrini@dima.unige.it University of Genova, Italy

Gert-Martin GREUEL greuel@mathematik.uni-kl.de University of Kaiserslautern, Germany

Klaus HULEK hulek@math.uni-hannover.de University of Hannover, Germany

ElhamIZADI izadi@math.uga.edu University of Georgia, U.S.A.

Vassil KANEV kanev@math.unipa.it University of Palermo, Italy

Victor KULIKOV kulikov@mi.ras.ru Steklov Institute, Moscow, Russia

Adrian LANGER langet@maths.warwick.ac.uk University of Warwick, U.K.

Vladimir LIN vlin@techunix.technion.ac.il Technion - Israel Inst. of Technology

Massimiliano MELLA mll@dns.unife.it University of Ferrara, Italy

Ferruccio ORECCHIA orecchia@unina.it Univ. of Naples "Federico II", Italy

Thomas PETERNELL Thomas.Peternell@uni-bayreuth.de University of Bayreuth, Germany

LIST OF PARTICIPANTS XI

Inna SCHERBAK scherbak@post.tau.ac.il Tel Aviv University, Israel

Edoardo SERNESI sernesi@mat.uniroma3.it University of Rome 3, Italy

Andrew J. SOMMESE sommese@nd.edu University of Notre Dame, U.S.A.

Peter TANNENBAUM peter _tannenbaum@csufresno.edu California State University, U.S.A.

Francesca TOVENA tovena@mat.uniroma2.it Univ. of Rome "Tor Vergata", Italy

Jonathan M. WAHL jmwahl@unc.edu University of North Carolina, U.S.A.

Shira ZUR zur@macs.biu.ac.il Bar-Han University, Israel

Rene SCHOOF schoof@mat.uniroma2.it Univ. of Rome "Tor Vergata", Italy

Eugenii SHUSTIN shustin0:}post.tau.ac.il Tel Aviv University, Israel

Balazs SZENDROI balazs@maths.warwick.ac.uk University of Warwick, U.K.

Mina TEICHER teicher@macs.biu.ac.il Bar-Han University, Israel

Mohammed ULUDAG uludam@macs.biu.ac.il Bar-Han University, Israel

Bronek WAJNRYB wajnryb@techunix.technion.ac.il Technion - Israel Inst. of Technology

LIST OF CONTRIBUTORS

Igor l. BURBAN burban@mathematik.uni-kl.de Fachbereich Mathematik UniversiHit Kaiserslautern Erwin-Schrodinger-StraBe 0-67663 Kaiserslautern. Germany

Luca CHIANTINI chiantini@unisi.it Dipartimento di Matematica Universita di Siena Via de Capitano 15 1-53100 Siena, Italy

Francesca CIOFFI cioffifr@unina.it Dip. di Matematica e Applicazioni Universita di Napoli "Federico II" Complesso di Monte S. Angelo Via Cintia 1-80126 Napoli, Italy

Yuriy DROZD yuriy@drozd.org Dept. of Mechanics and Mathematics Kyiv Taras Shevchenko University Volodimirska 64 252033 Kyiv, Ukraine

Alberto CALABRI calabri@mat.uniroma2.it Oipartimento di Matematica Universita di Roma "Tor Vergata" Via della Ricerca Scientifica 1-00133 Roma, Italy

Ciro CILIBERTO cilibert@mat.uniroma2.it Oipartimento di Matematica Universita di Roma "Tor Vergata" Via della Ricerca Scientifica 1-00133 Roma, Italy

Wolfram DECKER decker@math.uni-sb.de Fachrichtung Mathematik Universitat des Saarlandes 66041 Saarbrucken GERMANY

Michael FRYERS fryers@math.uni-hannover.de Institut fur Mathematik Universitat Hannover 0-30167 Hannover, Germany

XIII

XlV LIST OF CONTRIBUTORS

Gerard van der GEER Gert-Martin GREUEL geer@science.uva.nl Dept. of Math. and Computer Science University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands

Klaus HULEK hulek@math.uni-hannover.de lnstitut rur Mathematik Universitat Hannover 0-30167 Hannover, Germany

Jeremy Y. KAMINSKI kaminsj@macs.biu.ac.il Dept. of Math. and Computer Science Bar-Han University Ramat Gan 52900, Israel

Ron LIVNE rlivne@math.huji.ac.il Mathematics Institute The Hebrew University of Jerusalem Givat-Ram, Jerusalem 91904, Israel

Rick MIRANDA miranda@math.colostate.edu Department of Mathematics Colorado State University Fort Collins, CO 80523-1874, U.S.A.

Thomas PETERNELL Thomas.Petemell@uni-bayreuth.de Mathematisches lnstitut Universitat Bayreuth 0-95440 Bayreuth, Germany

greuel@mathematik.uni-kl.de Fachbereich Mathematik Universitat Kaiserslautem Erwin-Schrodinger-StraJ3e 0-67663 Kaiserslautern, Gemlany

Elham IZADI izadi@math.uga.edu Department of Mathematics University of Georgia Athens, GA 30602-7403 U.S.A.

Anatoly LIBGOBER libgober@math.uic.edu Department of Mathematics University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045, U.S.A.

Christoph LOSSEN lossen@mathematik.uni-kl.de Fachbereich Mathematik Universitat Kaiserslautem Erwin-Schrodinger-StraJ3e 0-67663 Kaiserslautem, Germany

Ferruccio ORECCHIA orecchia@unina.it Dip. di Matematica e Applicazioni Universita di Napoli "Federico II" Complesso di Monte S. Angelo Via Cintia 1-80126 Napoli, Italy

Rene SCHOOF schoof@mat.uniroma2.it Dipartimento di Matematica Universita di Roma "Tor Vergata" Via della Ricerca Scientifica 1-00133 Roma, Italy

LIST OF CONTRIBUTORS xv

Mathias SCHULZE mschulze@mathematik.uni-kl.de Fachbereich Mathematik Universitat Kaiserslautern Erwin-Schrodinger-StraBe D-67663 Kaiserslautern, Germany

Jeroen SPANDAW spandaw@math.uni-harillover.de Institut fur Mathematik Universitat Hannover D-30167 Hannover, Germany

Mina TEICHER teicher@macs.biu.ac.il Dept. of Math. and Computer Science Bar-Ilan University Ramat Gan 52900, Israel

Charles W. WAMPLER Charles. W. Wampler@gm.com General Motors Research Laboratories Mail Code 480-106-359 30500 Mound Road Warren, MI 48090-9055, U.S.A.

Andrew J. SOMMESE sommese@nd.edu http://www.nd.edu/ '"'-'sommese Department of Mathematics University of Notre Dame Notre Dame, IN 46556, U.S.A.

Balazs SZENDROI balazs@maths.warwick.ac.uk Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom

Jan VERSCHELDE jan@math.uic.edu http://www.math.uic.edul'"'-'jan Department of Mathematics University of Illinois at Chicago 851 South Morgan (M/C 249) Chicago, IL 60607-7045, U.S.A.