Commutative Algebra, Singularities and Computer Algebra: Proceedings of the NATO Advanced Research Workshop on Commutative Algebra, Singularities and Computer Algebra Sinaia, Romania 17–22 September 2002

274 Pages·2003·36.44 MB·English

by  Ciprian S. Borcea (auth.)

Commutative Algebra, Singularities and Computer
Algebra
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Series II: Mathematics, Physics and Chemistry - Vol. 115
Commutative Algebra,
Singularities and Computer
Algebra
edited by
Jurgen Herzog
Universităt Essen,
Fachbereich Mathematik und In formatik,
Essen, Germany
and
Victor Vuletescu
University of Bucharest,
Faculty of Mathematics and Informatics,
Bucharest, Romania
Springer-Science+Business Media, B.V.
Published in cooperation with NArO Scientific Affairs Division
Proceedings of the NATO Advanced Research Workshop on
Commutative Algebra, Singularities and Computer Algebra
Sinaia, Romania
17-22 September 2002
A C.I. P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-1487-1 ISBN 978-94-007-1092-4 (eBook)
DOI 10.1007/978-94-007-1092-4
Printed on acid-free paper
AII Rights Reserved
© 2003 Springer Science+Business Media Dordrecht
Softcover reprint of the hardcover 1s t edition 2003
Originally published by Kluwer Academic Publisher
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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Contents
Preface vii
List of Participants
IX
Ciprian Borceal Association for Flag Configurations
Winfried Bmns and Aldo Concal Grobner bases and determinantal ideals 9
Marc Chardin! Bounds for Castelnuovo-Mumford regularity in terms of
degrees of defining equations 67
Jorgen Backelin, Svetlana Cojocaru and Victor Ufnarovski/ The
Computer Algebra Package Bergman: Current State 75
Steven Dale Cutkosky! Monomialization and ramification of valuations 101
Alexandm Dimcal Hyperplane arrangements, M-tame polynomials and
twisted cohomology 113
David Eisenbud. Klaus Hulek and Sorin Popescu! A note on the
Intersection of Veronese Surfaces 127
Viviana Ene and Dorin Popescu! Rank one Maximal Cohen-Macaulay
Y;
modules over singularities of type r? + tl + ri + 141
Shiro Goto, Futosbi Hayasaka and Satoe Kasugal Towards a theory of
Gorenstcin m-primary integrally closed ideals 159
Hidefumi Ohsugi, Tomonori Kitamura and Takayuki Hibi/ Universal
Grobner bases, integer programming and finite graphs 179
Cristodor lonescul Torsion in tensor powers and flatness 191
Martin Kreuzer and Lorenzo Robbiano! Basic Tools for Computing in
Multigraded Rings 197
Gerbard pfister! A Problem in Group Theory solved by Computer
Algebra 217
Peter Schenzcl! On curves of small degree on a normal rational surface
scroll 225
Anna Torstensson,Viclor Ufnarovski and Hans Ofverbeckf On SAGS!
Bases and Resultants 241
Yuji Yoshino! Modules of G-dimension zero over local rings with the
cube of maximal ideal being zero 255
PREFACE
A NATO-Advanced Research Workshop "Commutative Algebra, Singulari
ties and Computational Algebra" was held in Sinaia, Romania, from 17 to 22
September2002. The financial support wasofTered by the NATO grant PST. ARW.
978769. The Workshop was organized by Jiirgen Herzog and Victor Vulctescu
with the assistance of the members of the Organizing Committee: Alexandru
Dimca, David Eisenbud, Dorin Popescu and Lorenzo Robbiano. There were 27
lectures given by the participants and open discussions on the subjects presented.
The Scientific Program highlighted current trends in the development ofCommu
talive Algebra, Singularities Theory and their computational aspects. There were
also reports on the present status of the computer algebra packages Bergmann, Co
CoA and SINGULAR, and of various applications of them. The volume contains
research papers and also surveys reflecting the topics discussed in the lectures. It
is directed to researchers and graduate students and is intended to stimulate further
research in these topics. The editors would like to thank the contributors of this
volume and the NATO Scientific Division for their generous financial support
January 2003.
Jiirgen Herzog,
Victor Vuletescu
vii
NATO-Advanced Research Workshop
"Commutative Algebra, Singularities and Computer
algebra"
Sinaia, Romania, 17-22 September, 2002.
List of participants
Aneta Aramova, University of Sofia, Faculty of Mathematics and Informa
tics, James Bourchier Blvd. 5, Sofia, Bulgaria.
Lucian Biidescu, University of Bucharest, Faculty of Mathematics,
Academiei st 14, Bucharest, Romania.
Mircea Beeheanu, University of Bucharest, Faculty of Mathematics,
Academiei .'It 14, Bucharest, Romania.
Gabriele Biucchi, Humboldt University of Berlin, Unter den Linden 6, Berlin,
Germany.
Andrei Bruicov, State University of Tiraspol, Chiilinau, Moldavia.
Giprian Boreca, Rider University, Department of Mathematics,
Lawrenceville Road 2083, Lawrenceville, USA.
Winfired Bruns, University of Osnabrueck, Albrechtstr. 28, Osnabrueck,
Germany.
Mare Gllardin, Institut Mathematique de Jussieu, Universite Piere et Marie
Curie, Place de Jussieu 4, Paris, France.
Svetlana Cojocaru, Institute of Mathematics and Computer Science,
Academiei st 5, Chi&inau, Moldavia.
Aldo Conca, University of Genova, Via Dodecaneso 35, Genova, Italy.
Steven-Dale Cutkosky, University of Missouri Columbia, MO 65221, USA.
Alexandru Dimea, University of Bordeaux, Cours de Liberation 351,
Bordeaux, France.
Viviana Ene, University Ovidius, Bd. Mamaia 124, Constanta, Romania.
Shiro Coio, Department of Mathematics, Meiji Ulliversity,
1-1-1 Higa.~himita, Tama-ku, Kawasaki, 214-8571, Japan.
Jiirgen Herzog, University of Essen, Universitiitstrasse 3, Essen, Germany.
Takayuki J/ibi, Department of Mathematics, Graduate School of Science,
Osaka University, Toyonaka, Osaka, 560-0043, Japan.
Cristodor [oneseu, Institute for Mathematics of The Romanian Academy,
Bucharest, Romania.
Martin Kreuzer, University of Regensbnrg, Universitaatstra.~se 31, Regens
burg, Germany.
ix
Gerhard Pfister, University of Kaiserslautern, Kaiserslautcrn, Germany.
Winfried Pohl, University of Kaiserslautern, Kaiscrslautern, Germany.
Mihail Popa, Institute of Mathematics am.! Computer Science, Moldova,
Academiei st,5, Chi~inau, Moldova.
Valeriu Popa, Institute of Mathematics and Computer Science, Moldova,
Academiei sl,5, Chi~inau, Moldova.
Dorin Popescu, University of Bucharest, Faculty of Mathematics, Academiei
st 14, Bucharest, Romania.
Sorin Popescu, University of Stony Brook, Stony Brook, NY 11794-3651,
USA.
Lorenzo Robbiano, University of Genova, Via Dodecaneso 35, Genova, Italy.
Maria-Evelina Rossi, University of Genova, Via Dodecaneso 35, Genova,
Italy.
Peter Schenzel, University of Halle, Institllt fiir Informatik, Von-Seckcndorff
Pl. 1 Halle, Germany.
Victor Ufnarovski, Institute of Mathematics and Computer Science,
Moldova, Academiei sl,5, Chisinau, Moldova.
Crlstia1l Vuica, University of Bucharest, Faculty of Mathematics,
Academiei st 14, Bucharest, Romania
Victor Vuletescu, University of Bucharest, Faculty of Mathematics,
Academiei st 14, Bucharest, Romania.
Yuji Yoshino, Department of Mathematics, Faculty of Science, Okayama
University, Okayama, 700-8530, Japan.
Santiago Zarzuela, University of Barcelona, Gran Via 585, Barcelona, Spain.
x
ASSOCIATION FOR FLAG CONFIGURATIONS
CIPRIAN S. BORCEA
Rider University
Lawrenceville. NJ 08648 u.s.A.
(borcea~rider.edu)
Ab5traCt. We extend the classical ootion of association from point configurations in projective
spaces to Hag configurations.
Key words: association, point configurlitioIl'l, flag oonfigUIlltions.
AMS Subject Classification: 14NlS, 14E05, 14L30, }4N20
Introduction
The notion of association, as used for example by Castelnuovo or Coble [2J.
expresses the fact that the projective invariants of configurations of n ordered
points in IJ» D~l can be identified with the projective invariants of associated con
figurations of n points in IF',,_D_I'
This duality manifests tiself in various ways, and one may find modern consider
ations related to it in [31. [4J, [5]. [7].
OUf purpose here is to extend the notion of association from point configura
tions to flag configurations. For simplicity, we consider the ground field to be the
complex numbers C.
Let.'Fp = GL(D)/P = F(d , ..., dsID) denote the flag variety consisting of (partial)
l
flags:
V :0= Vo C VI C V2 C ... C Vs ccP = Vs+1 ' dimc(VIJVk_l) = dk
In the homogeneous space description GL(D)/P, P denotes the stabilizer of a
reference flag as above: a parabolic subgroup in the general linear group GL(D).
We consider oonfigurations of n points in.'Fp up to equivalence under GL(D). This
orbit space must be adapted somehow before converting into ap rojective algebraic
variety, but its birational type is well defined [8], and all our considerations will
J. Hen.og and V. Vulelucu (eds), COmnuJuuivt: Algebro. Singularilies and Computer Algebra, 1-8.
0 2003 Kluwer Academic Publishers.
be up to biraJiQnaI equivalence. Thus, our quotient notation for the configuration
space:
F(d, , . .., d,ID)" / /GL(D) ~ F.(dw.,d,ID)
refers to any birational model.
Extending association to flag configurations means finding (for adequate dimen
sions d and 0) a nalUral biraJionai equivalence:
k
F(d, , ... ,d,ID)" / /GL(D)· · -> F(J" ... ,J,lii)" / /GL(ii)
which is involutive, and reduces to association for point configurations when s = I
and d I = 1. This is obtained in our Theorem 3.2.
1. Association for point configurations
We give here a brief description of the classical notion of association. It refers to
general configurations of n > D + I ordered points in lP'D_\, respectively lP',,_o_J>
that is:
x= (xI' ... 'x,,) E (lP'O_I)" and x= (xII ... ,x,,) E (lP',,_O_I)"
The assumption that x, respectively X, arc general means that their orbits under
GL(D). respectively GL(n - D). are of maximal dimension.
Let X be a D x n matrix made of columns representing the points xl, ... ,xlI' and
similarly X a matrix (n - D) x n, made of columns representing XI' ... ,x".
The two configurations are called associated when the relation:
is satisfied for somc invenible diagonal n x n matrix T.
A more invariant way to say the same is the following. Let:
lP'D_I X Jil'1I_D_1 --t lP'D{1I_0)_1' (u, v) I-jo u ® v
denote the Scgrc embedding. Then x and i are arsocialed when the images xk ®
ik, k = I I .•. ,n span an (n - 2) projective subspace. This means that a dependency
occurs by comparison with the general case of unrelated x and y.
Example: A general configuration of six points in lP'z becomes self-arsociated
when the six points lie on a conic.
Indeed, the relevant part of the Segre embedding reduces in this case to the qua
dratic Veronese: Jil'z --t lP'~. and the images of six points span less than IP'~ only
when the points lie on a conic.
2