CONTEMPORARY MATHEMATICS

331

Commutative Algebra Interactions with Algebraic Geometry

International Conference Grenoble, France

July 9-13, 2001

Special Session at the Joint International Meeting of the American Mathematical Society

and the Societe Mathematique de France Lyon, France

July 17-20, 2001

Luchezar L. Avramov Marc Chardin

Marcel Morales Claudia Polini

Editors

Commutative Algebra Interactions with Algebraic Geometry

http://dx.doi.org/10.1090/conm/331

CoNTEMPORARY MATHEMATICS

331

Commutative Algebra Interactions with Algebraic Geometry

International Conference Grenoble, France

July 9-13, 2001

Special Session at the Joint International Meeting of the American Mathematical Society

and the Societe Mathematique de France Lyon, France

July 17-20, 2001

Luchezar L. Avramov Marc Chardin

Marcel Morales Claudia Polini

Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Dennis DeThrck, managing editor

Andreas Blass Andy R. Magid Michael Vogelius

This volume contains articles based on invited talks given at two international confer-ences held in France in 2001.

2000 Mathematics Subject Classification. Primary 13B22, 13C40, 13D02, 13D45, 13E15, 13H10, 13H15, 14E22, 14M05, 35Nl0.

Library of Congress Cataloging-in-Publication Data Commutative algebra : interactions with algebraic geometry : international conference on algebre commutative, interactions avec Ia geometrie algebrique, July 9~ 13, 2001, and Special Session on Commutative Algebra and its Interactions with Algebraic Geometry, July 17~20, 2001 / Luchezar L. Avramov ... [eta!.], editors.

p. em. -(Contemporary mathematics ; 331) Includes bibliographical references. ISBN 0-8218-3233-6 (alk. paper) 1. Commutative algebra-Congresses. I. Avramov, L. L. (Luchezar L.), 1948~ II. Special

Session on Commutative Algebra and its Interactions with Algebraic Geometry (2001 : Lyon, France) III. Contemporary mathematics (American Mathematical Society) ; v. 331.

QA251.3.C654 2003 5121.24--dc21 2003050343

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Contents

Introduction vii

List of talks and references to publications 1x

Linearization of local cohomology modules JOSEP ALVAREZ MONTANER AND SANTIAGO ZARZUELA 1

Multiplicity-free subvarieties of flag varieties MICHEL BRION 13

Morita contexts, idempotents, and Hochschild cohomology-with applications to invariant rings-RAGNAR-OLAF BUCHWEITZ 25

Some applications of two dimensional complete ideals ANTONIO CAMPILLO AND SANTIAGO ENCINAS 55

Generically finite morphisms and simultaneous resolution of singularities STEVEN DALE CUTKOSKY 75

Two results on the number of generators JUAN ELIAS 101

When is a binomial ideal equal to a lattice ideal up to radical? KAZUFUMI ETO 111

Depth and amplitude for unbounded complexes HANS-BJ0RN FOXBY AND SRIKANTH IYENGAR 119

On the ideal of minors of matrices of linear forms ANNA GUERRIERI AND IRENA SWANSON 139

Surjectivity of multiplication and F-regularity of multigraded rings MITSUYASU HASHIMOTO 153

On the Ext-modules of ideals of Borel type JURGEN HERZOG, DORIN POPESCU, AND MARIUS VLADOIU 171

Sums of linked tepli ideals MARK R. JOHNSON 187

Cartan involutiveness = Mumford regularity BERNARD MALGRANGE 193

v

vi CONTENTS

The Frobenius endomorphism and homological dimensions CLAUDIA MILLER 207

Characterization of some projective subschemes by locally free resolutions UWE NAGEL 235

On unique factorization in semigroups of complete ideals OLIVIER PILTANT 267

Modules with strange homological properties and Chebychev polynomials JAN-ERIK Roos 295

Notes on the behavior of the Ratliff-Rush filtration MARIA EVELINA ROSSI AND IRENA SWANSON 313

On symbolic powers of prime ideals SEAN SATHER-VVAGSTAFF 329

Multiplicities and the number of generators of Cohen-Macaulay ideals VVOLMER V. VASCONCELOS 343

Chains of integrally closed ideals KEI-lCHI VVATANABE 353

Introduction

The authors of the papers collected in this volume gave invited talks at one of the following meetings, held in France during the summer of 2001:

• Algebre commutative. Interactions avec la geometrie algebrique, Institut Fourier, Grenoble, July g_:_July 13, 2001;

• Commutative Algebra and its Interactions with Algebraic Geometry, Special Session at the Joint meeting of the American Mathematical Society and the Societe Mathematique de France, Lyon, July 17-July 20, 2001.

The organizers (L. A. and M. M. for Grenoble, M. C. and C. P. for Lyon) planned the meetings in close collaboration. A significant overlap of participants was expected, so duplication of lectures was avoided. A list of all the talks given at each meeting is included following this introduction. The speakers have provided references to the articles where the results presented at the meetings have appeared or will appear in print.

The Centre National de la Recherche Scientifique, the lnstitut Fourier, the Ministere des Affaires Etrangeres, the Ministere de la Recherche, and the Societe Mathematique de France provided partial financial support to one or both meetings. It is gratefully acknowledged. The conferences were held at the lnstitut Fourier in Grenoble and the Ecole Normale Superieure in Lyon. We thank these institutions for their hospitality, their staff for considerable help in organizing and running the meetings, and Mrs. Arlette Guttin-Lombard from the Institut Fourier for preparing the final typescripts of many papers.

This volume owes a lot to the input of the participants in the meetings, the speakers in Grenoble and Lyon, the authors who submitted papers to these pro-ceedings, and the experts who refereed the manuscripts. We extend our thanks to all of them.

Luchezar Avramov Marc Chardin Marcel Morales Claudia Polini

L. A. and C. P. were partly supported by grants from the N.S.F. vii

List of Talks and References to Publications

Grenoble

Joseph P. Brennan (North Dakota State University, Fargo, ND, U.S.A.) Prolongation of invariants [7]

Michel Brion (University of Grenoble, France) Positivity in the Grothendieck group of flag varieties [9]

Winfried Bruns (University of Osnabriick, Germany) Polyhedral K2 [11]

Ragnar-Olaf Buchweitz (University of Toronto, Canada) Hochschild cohomology and the centre of the derived category -The case of Kleinian singularities

Marc Chardin (University of Paris VI, France) Cohomology of projective schemes: Prom annihilators to vanishing [14]

Aldo Conca (University of Genoa, Italy) Algebras of minors [10]

Steven Dale Cutkosky (University of Missouri, Columbia, MO, U.S.A.) Ramification of valuations

Felix Delgado de la Mata (University of Valladolid, Spain) Poincare series for curve singularities and Alexander polynomial [15]

Jean-Pierre Demailly (University of Grenoble, France) Subadditivity of multiplier ideal sheaves and asymptotic Zariski decomposition [21], [22]

Juan Elias (University of Barcelona, Spain) On the Grauert-Riemenschneider vanishing theorem [23]

Santiago Encinas (University of Valladolid, Spain) Algorithmic equiresolution [24]

Kazufumi Eto (Nippon Institute of Technology, Saitama, Japan) Almost complete intersection lattice ideals [25]

Hubert Flenner (Ruhr-Universitat, Bochum, Germany) Rational curves and rational singularities [26]

Hans-Bj0rn Foxby (University of Copenhagen, Denmark) Duality and equivalence [2], [28]

Ralf Froberg (University of Stockholm, Sweden) On the number of ideals of finite colength [6]

ix

X LIST OF TALKS AND REFERENCES TO PUBLICATIONS

Alexander Gerko (Moscow State University, Russia) Homological dimensions [31], [32]

Joseph Gubeladze (A. Razmadze Mathematical Institute, Tbilisi, Georgia) Higher K-theory of toric varieties [30]

Mitsuyasu Hashimoto (Nagoya University, Japan) F -regularity of multi-graded rings

Jiirgen Herzog (University of Essen, Germany) Sequentially Cohen-Macaulay modules and local cohomology

Srikanth Iyengar (University of Missouri, Columbia, MO, U.S.A.)

[38]

[39]

Andre-Quillen homology of algebra retracts [3] Clas Lofwall (University of Stockholm, Sweden)

Homology of commutative rings defined by many generic quadratic relations

Gennady Lyubeznik (University of Minnesota, Minneapolis, MN, U.S.A.) A criterion for the vanishing of local cohomology modules in characteristic p

Bernard Malgrange (University of Grenoble, France) On differential Galois theory for non-linear equations [42], [43]

Claudia Miller (Syracuse University, Syracuse, NY, U.S.A.) Probenius Powers of complete intersections [4], [44]

Uwe Nagel (University of Kentucky, Lexington, KY, U.S.A.) Double lines and their even liaison classes [45]

Le Thi Thanh Nhan (Thai Nguyen University, Vietnam) On Noetherian dimension of Artinian modules [20]

Claudia Polini (University of Notre Dame, South Bend, IN, U.S.A.) Integral closure of ideals and Cohen-Macaulayness of special fiber rings [16]

Dorin Popescu (Institute of Mathematics, Bucharest, Romania) Cohen-Macaulay representation on cubic surfaces [41]

Sorin Popescu (State University of New York, Stony Brook, NY, U.S.A.) Exterior algebra methods for syzygies

Ana Reguera (University of Valladolid, Spain) Divisorial valuations dominating a normal surface singularity [19]

Jan-Erik Roos (University of Stockholm, Sweden) Theorems and conjectures in homological algebra inspired by computer calculations [49], [50]

Maria Evelina Rossi (University of Genoa, Italy) The Castelnuovo-Mumford regularity of the tangent cone of a local ring [51]

Sean Sather-Wagstaff (University of Illinois at Urbana-Champaign, IL, U.S.A.) Intersections of symbolic powers of prime ideals [52], [53]

Peter Schenzel (Martin-Luther University Halle-Wittenberg, Halle, Germany) On curves of small degree on a rational normal scroll [54]

Aron Simis (Federal University of Pernambuco, Brazil) Remarks on Cremona transformations [57]

LIST OF TALKS AND REFERENCES TO PUBLICATIONS

Apostolos Thoma (University of Ioannina, Greece) On the bounds of the binomial arithmetical rank for simplicial toric varieties

Giuseppe Valla (University of Genoa, Italy) Castelnuovo regularity and finiteness of Hilbert functions

Wolmer V. Vasconcelos (Rutgers University, Trenton, NJ, U.S.A.) Integral Closure: Finding it and bounding it

Oana Veliche (Purdue University, West Lafayette, IN, U.S.A.) Finite homological dimensions

Alberto Vigneron-Tenorio (University of Sevilla, Spain) Regularity of a homogeneous lattice ideal

Kei-ichi Watanabe (Nihon University, Tokyo, Japan) F-rationality and F-regularity of Rees algebras

Santiago Zarzuela (University of Barcelona, Spain) Betti numbers of arrangements of linear varieties and local cohomology

Lyon

Margherita Barile (University of Bari, Italy) Effective construction of irreducible curve singularities

Winfried Bruns (University of Osnabriick, Germany)

xi

[59]

[60]

[8]

[35], [36]

[1]

Minors of symmetric and exterior powers [12] Ragnar-Olaf Buchweitz (University of Toronto, Canada)

Hochschild cohomology and idempotents [13] Alberto Corso (University of Kentucky, Lexington, KY, U.S.A.)

Generalized Sally modules [19] Vincent Cossart (University of Versailles, France)

Tours de Semple et espaces de Gruson-Belghitti [18] Stephen Dale Cutkosky (University of Missouri, Columbia, MO, U.S.A.)

Hilbert functions of ideal sheaves Mireille Martin-Deschamps (University of Versailles, France)

Triades et familles de courbes [37] Vesselin Gasharov (Purdue University, West Lafayette, IN, U.S.A.)

Stable ideals in polynomial rings with restricted powers of the variables [29]

Laura Ghezzi (University of Missouri, Columbia, MO, U.S.A.) On the depth of the associated graded ring of an ideal [33]

Anna Guerrieri (University of L'Aquila, Italy) On the primary decomposition of the ideals generated by the minors of 1-generic matrices [34]

Mark Johnson (University of Arkansas, Fayetteville, AK, U.S.A.) Normality of Rees algebras of modules having second analytic deviation one

Monique Lejeune-Jalabert (University of Versailles, France) Valuations in functions fields of surfaces and toric geometry [40]

xii LIST OF TALKS AND REFERENCES TO PUBLICATIONS

Marcel Morales (University of Grenoble, France) Set-theoretic complete intersection on binomials [5]

Irena Peeva (Cornell University, Ithaca, NY, U.S.A.) Connectedness of Hilbert schemes [46]

Marc Perret (Ecole Normale Superieure, Lyon, France) Sur le nombre de points des varietes unirationnelles de dimension 3 sur un corps fini [47]

Olivier Piltant (University of Versailles, France) On semigroups of complete ideals having unique factorization [48]

Sorin Popescu (State University of New York, Stony Brook, NY, U.S.A.) Exterior algebra methods for syzygies

Maria Evelina Rossi (University of Genoa, Italy) Maximal Hilbert finctions of a local ring [51]

Aron Simis (Federal University of Pernambuco, Brazil) On derivations and tangent vector fields

Irena Swanson (New Mexico State University, Las Cruces, NM, U.S.A.) Generic freeness via prime sum filtrations

Bernd Ulrich (Purdue University, West Lafayette, IN, U.S.A.) Integral extensions and multiplicities [56]

Wolmer V. Vasconcelos (Rutgers University, Trenton, NJ, U.S.A.) Multiplicity and reduction numbers of ideals [58]

Jie-Tai Yu (The University of Hong Kong, China) Coordinates of polynomial algebras

References [1] J. Alvarez-Montaner, R. Garca-L6pez, S. Zarzuela, Betti numbers of arrangements of linear

varieties and local cohomology, Adv. Math., 174 (2003), 35-56. [2] L. L. Avramov, H.-B. Foxby, Ring homomorphisms and finite Gorenstein dimension, Proc.

London Math. Soc. (3) 75 (1997), 241-270. [3] L. L. Avramov, S. Iyengar, Andre- Quillen homology of algebra retracts, Ann. Sci. Ecole Norm.

Sup. (4), to appear. [4] L. L. Avramov, C. Miller, Frobenius powers of complete intersections, Math. Res. Lett. 8

(2001), 225-232. [5] M. Barile, M. Morales, A. Thoma, Set-theoretic complete intersections on binomials, Proc.

Amer. Math. Soc. 130 (2002), 1893-1903. [6] V. Barucci, R. Froberg, On the number of ideals of finite colength, Geometric and Combi-

natorial Aspects of Commutative Algebra (Messina, 1999), Lecture Notes in Pure and Appl. Math. 217, Dekker, New York, 2001; pp. 11-19.

[7] J. P. Brennan, Prolongation of invariants, in preparation. [8] E. Briales, A. Campillo, P. Pis6n, A. Vigneron, Simplicial complexes and syzygies of lattice

ideals, Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000), Contemp. Math. 286, Amer. Math. Soc., Providence, RI, 2001; pp. 169-183.

[9] M. Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), 137-159.

[10] W. Bruns, A. Conca, Algebras of minors, J. Algebra 246 (2001), 311-330. [11] W. Bruns, J. Gubeladze, Polyhedral K2, manuscripta math. 109 (2002), 367-404. [12] W. Bruns, W. V. Vasconcelos, Minors of symmetric and exterior powers, J. Pure Appl.

Algebra, to appear. [13] R.-0. Buchweitz, Morita contexts, idempotents, and Hochschild cohomology with applications

to invariant rings, these Proceedings.

LIST OF TALKS AND REFERENCES TO PUBLICATIONS xiii

[14] Marc Chardin, Cohomology of projective schemes: From annihilators to vanishing, math.AC/0209338.

[15] A. Campillo, F. Delgado, S.M. Gusein-Zade, The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math. J., to appear.

[16] A. Corso, L. Ghezzi, C. Polini, B. Ulrich, Cohen-Macaulayness of special fiber rings, Comm. Algebra (volume dedicated to S. Kleiman), to appear.

[17] A. Corso, C. Polini, M. Rossi, Depth of associated graded rings via Hilbert coefficients of ideals, submitted.

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[20] N. T. Cuong, M. Morales, L. T. Nhan, On the length of generalized fractions, J. Algebra, to appear.

[21] J.-P. Demailly, L. Ein, R. Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. (volume in honor of William Fulton), 48 (2000), 137-156.

[22] J.-P. Demailly, Th. Peternell, M. Schneider, Pseudo-effective line bundles on compact Kahler manifolds, lnternat. J. Math, 6 (2001), 689-741

[23] J. Elias, On the Grauert-Riemenschneider vanishing theorem, J. of Algebra 248 (2002), 265-271.

[24] S. Encinas, A. Nobile, 0. Villamayor, On algorithmic equiresolution and stratification of Hilbert schemes, Proc. London Math. Soc, to appear (http://arXiv.org/abs/math.AG/0010228).

[25] K. Eto, Almost complete intersection lattice ideals, submitted.

[26] H. Flenner, M. Zaidenberg, Rational curves and rational singularities., Math. Z., to appear. [27] H.-B. Foxby and S. Iyengar, Depth and amplitude for unbounded complexes, these Proceed-

ings. [28] A. Frankild, P. J0rgensen, Foxby equivalence, complete modules, and torsion modules, J. Pure

Appl. Algebra 174 (2002), 135-147. [29] V. Gasharov, T. Hibi, I. Peeva, Resolutions of a-stable ideals, J. Algebra 254 (2002), 375-394. [30] J. Gubeladze, Higher K-theory of toric varieties, http: I /arxiv. org/abs/math. KT/0104166 [31] A. Gerko, On homological dimensions, Sb. Math. 192 (2001), 1165-1179. [32] A. Gerko, On suitable modules and G-perfect ideals, Russ. Math. Surv. 56 (2001), 749-750. [33] L. Ghezzi, On the depth of the associated graded ring of an ideal, J. Algebra 248 (2002),

688-707. [34] A. Guerrieri, I. Swanson, On the ideal of minors of matrices of linear forms, these Proceed-

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153-190. [36] N. Hara, K. Watanabe, K. Yoshida, Rees algebras ofF-regular type, J. of Alg. 247 (2002),

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[45] U. Nagel, R. Notari, M. L. Spreafico, Curves of degree two and ropes on a line: their ideals and even liaison classes, J. Algebra, to appear.

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Algebra (volume dedicated to Walmer V. Vasconcelos), to appear. [51] M. E. Rossi, N. V. Trung, G. Valla, Castelnuovo-Mumford regularity and extended degree,

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65 (2002), 56Q-574. [53] S. Sather-Wagstaff, On symbolic powers of prime ideals, these Proceedings. [54] P. Schenzel, On curves of small degree on a rational normal scroll, preprint, Martin-Luther

University Halle-Wittenberg, Halle, 2002. [55] A. Sirois, On derivations and tangent vector fields, in preparation. [56] A. Sirois, B. Ulrich, W. V. Vasconcelos, Codimension, multiplicity and integral extensions,

Math. Proc. Cambridge Philos. Soc. 130 (2001), 237-257. [57] A. Sirois, R. Villarreal, Constraints for the normality of monomial subrings and birationality,

Proc. AMS, to appear (electronically published on November 13, 2002). [58] W. V. Vasconcelos, Multiplicities and reduction numbers, Compositio Math., to appear. [59] W. V. Vasconcelos, Integral closure, book in preparation. [60] 0. Veliche, Construction of modules with finite homological dimension, J. Algebra 250 (2002),

427-449.

Titles in This Series

331 Luchezar L. Avramov, Marc Chardin, Marcel Morales, and Claudia Polini, Editors, Commutative algebra: Interactions with algebraic geometry, 2003

330 S. Y. Cheng, C.-W. Shu, and T. Tang, Editors, Recent advances in scientific computing and partial differential equations, 2003

329 Zhangxin Chen, Roland Glowinski, and Kaitai Li, Editors, Current trends in scientific computing, 2003

328 Krzysztof Jarosz, Editor, Function spaces, 2003 327 Yulia Karpeshina, Giinter Stolz, Rudi Weikard, and Yanni Zeng, Editors,

Advances in differential equations and mathematical physics, 2003 326 Kenneth D. T-R McLaughlin and Xin Zhou, Editors, Recent developments in

integrable systems and Riemann-Hilbert problems, 2003 325 Seok-Jin Kang and Kyu-Hwan Lee, Editors, Combinatorial and geometric

representation theory, 2003 324 Caroline Grant Melles, Jean-Paul Brasselet, Gary Kennedy, Kristin Lauter,

and Lee McEwan, Editors, Topics in algebraic and noncommutative geometry, 2003 323 Vadim Olshevsky, Editor, Fast algorithms for structured matrices: theory and

applications, 2003 322 S. Dale Cutkosky, Dan Edidin, Zhenbo Qin, and Qi Zhang, Editors, Vector

bundles and representation theory, 2003 321 Anna Kaminska, Editor, Trends in Banach spaces and operator theory, 2003 320 William Beckner, Alexander Nagel, Andreas Seeger, and Hart F. Smith,

Editors, Harmonic analysis at Mount Holyoke, 2003 319 W. H. Schikhof, C. Perez-Garcia, and A. Escassut, Editors, Ultrametric functional

analysis, 2003 318 David E. Radford, Fernando J. 0. Souza, and David N. Yetter, Editors,

Diagrammatic morphisms and applications, 2003 317 Hui-Hsiung Kuo and Ambar N. Sengupta, Editors, Finite and infinite dimensional

analysis in honor of Leonard Gross, 2003 316 0. Cornea, G. Lupton, J. Oprea, and D. Tanre, Editors, Lusternik-Schnirelmann

category and related topics, 2002 315 Theodore Voronov, Editor, Quantization, Poisson brackets and beyond, 2002 314 A. J. Berrick, Man Chun Leung, and Xingwang Xu, Editors, Topology and

Geometry: Commemorating SISTAG, 2002 313 M. Zuhair Nashed and Otmar Scherzer, Editors, Inverse problems, image analysis,

and medical imaging, 2002 312 Aaron Bertram, James A. Carlson, and Holger Kley, Editors, Symposium in

honor of C. H. Clemens, 2002 311 Clifford J. Earle, William J. Harvey, and Sevin Recillas-Pishmish, Editors,

Complex manifolds and hyperbolic geometry, 2002 310 Alejandro Adem, Jack Morava, and Yongbin Ruan, Editors, Orbifolds in

mathematics and physics, 2002 309 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Integrable systems,

topology, and physics, 2002 308 Martin Guest, Reiko Miyaoka, and Yoshihiro Ohnita, Editors, Differentiable

geometry and integrable systems, 2002 307 Ricardo Weder, Pavel Exner, and Benoit Grebert, Editors, Mathematical results in

quantum mechanics, 2002 306 Xiaobing Feng and Tim P. Schulze, Editors, Recent advances in numerical methods

for partial differential equations and applications, 2002

TITLES IN THIS SERIES

305 Samuel J. Lomonaco, Jr. and Howard E. Brandt, Editors, Quantum computation and information, 2002

304 Jorge Alberto Calvo, Kenneth C. Millett, and Eric J. Rawdon, Editors, Physical knots: Knotting, linking, and folding geometric objects in JR3 , 2002

303 William Cherry and Chung-Chun Yang, Editors, Value distribution theory and complex dynamics, 2002

302 Yi Zhang, Editor, Logic and algebra, 2002 301 Jerry Bona, Roy Choudhury, and David Kaup, Editors, The legacy of the inverse

scattering transform in applied mathematics, 2002 300 Sergei Vostokov and Yuri Zarhin, Editors, Algebraic number theory and algebraic

geometry: Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday, 2002 299 George Kamberov, Peter Norman, Franz Pedit, and Ulrich Pinkall, Quaternions,

spinors, and surfaces, 2002 298 Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors,

Computational and statistical group theory, 2002 297 Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, and Brian

Parshall, Editors, Recent developments in infinite-dimensional Lie algebras and conformal field theory, 2002

296 Sean Cleary, Robert Gilman, Alexei G. Myasnikov, and Vladimir Shpilrain, Editors, Combinatorial and geometric group theory, 2002

295 Zhangxin Chen and Richard E. Ewing, Editors, Fluid flow and transport in porous media: Mathematical and numerical treatment, 2002

294 Robert Coquereaux, Ariel Garcia, and Roberto Trinchera, Editors, Quantum symmetries in theoretical physics and mathematics, 2002

293 Donald M. Davis, Jack Morava, Goro Nishida, W. Stephen Wilson, and Nobuaki Yagita, Editors, Recent progress in homotopy theory, 2002

292 A. Chenciner, R. Cushman, C. Robinson, and Z. Xia, Editors, Celestial Mechanics, 2002

291 Bruce C. Berndt and Ken Ono, Editors, q-series with applications to combinatorics, number theory, and physics, 2001

290 MichelL. Lapidus and Machiel van Frankenhuysen, Editors, Dynamical, spectral, and arithmetic zeta functions, 2001

289 Salvador Perez-Esteva and Carlos Villegas-Blas, Editors, Second summer school in analysis and mathematical physics: Topics in analysis: Harmonic, complex, nonlinear and quantization, 2001

288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001

287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001

286 Edward L. Green, Serkan Ho§ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001

285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001

284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001

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This volume contains 21 articles based on invited talks given at two international confer- ences held in France in 2001. Most of the papers are devoted to various problems of commutative algebra and their relation to properties of algebraic varieties. The book is suitable for graduate students and research mathematicians intmsted in commutative algebra and algebraic geometry.

ISBN 0-8218-3233-6

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