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Progress in Mathematics Volume 134
Series Editors
H. Bass J. Oesterle A. Weinstein
Algebraic Geometry and Singularities
Antonio Campillo L6pez Luis Narvaez Macarro Editors
Birkhauser Verlag Basel· Boston· Berlin
Editors:
Antonio Campillo LOpez Dep. de Algebra, Geometria y Topologia Fac. de Ciencias Universidad de Valladolid Prado de la Magdalena sIn 47005 Valladolid Spain
Luis Narvaez Macarro Dept. of Algebra, Computacion, Geometria y Topologia Fac. de Matematicas Universidad de Sevilla Tarfia sin 41012 Sevilla Spain
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Algebraic geometry and singularities I Antonio Campillo Lopez; Luis Narvaez Macarro (ed.). - Basel; Boston; Berlin: Birkhiiuser, 1996
(Progress in mathematics ; Vol. 134)
ISBN-13: 978-3-0348-9870-6 e-ISBN-13: 978-3-0348-9020-5 001: 10.1007/978-3-0348-9020-5
NE: Campillo Lopez, Antonio [Hrsg.]; GT 1991 Mathematics Subject Classification 14B05.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission of the copyright owner must be obtained.
@1996 Birkhiiuser Verlag, P.O. Box 133, CH-40l0 Basel, Switzerland Softcover reprint of the hardcover 18t edition 1996
Printed on acid-free paper produced of chlorine-free pulp. TCF 00
987654321
Table of Contents
Introduction ....
Plenary Conferences ..
Specialized Conferences
List of Participants . . .
I Resolution of Singularities
Desingularisation en dimension 3 et caracteristique p VINCENT COSSART
1 Differentes notions de desingularisation. . . _ . _ _ . _ _ 2 Premiere reduction . _ _ . _ . . _ _ . _ _ . . _ . _ _ . _ _
3 Deuxieme reduction, construction d'un modele projectif 4 Troisieme reduction, birationnel devient projectif 5 Final: Morphisme projectif birationnel devient
desingularisation . _ . . . . . . . . . . . . .
Sur l'espace des courbes tracees sur une singularite G. GONZALEZ-SPRINBERG M. LEJEC'NE-JALABERT
1 Introduction ................... . 2
3 4 5 6
Structure pro-algebrique de l'espace des courbes et la fonction de M. Artin d'une singularite . . . _ . . FamilIes de courbes (selon J. )Jash) et desingularisations Courbes sur une singularite isolee d'hypersurface Courbes lisses sur une singularite de surface Deux exemples . . . . . . . . . . . . . _ . . . . .
Blowing up acyclic graphs and geometrical configurations CARLOS MARIJUAN
1 Introduction..... _ . . . . 2 Basic concepts and notations 3 Blowing up acyclic graphs . . 4 Graphic representation of the blowing up for a
geometric configuration ..... . . . . . 5 Geometric modification for acyclic graphs _ . .
xi
xiii
xv
xix
3 4 4
5
5
9
10 13 17 22 27
33 34 37
42 46
vi Table of Contents
On a Newton polygon approach to the uniformization of singularities of characteristic p
T. T. l\loH 1 Introduction ..................... . 2 3 4
5 6
Newton polygon and uniformization for 7) ~ n - 1 Jumping lenuna and Vniformization for 7) = n - 2 The classification of 3-dimensional singularities and uniformization for A2 ~ 3 or A2 = 2, 7l"2 ~ 2 Uniformization for A2 = 2 and 7l"2 = 1 Uniformization for A2 = 1 ....... .
Geometry of plane curves via toroidal resolution MUTSUO OKA
49 53 57
64 74 81
1 Introduction.......................... 95 2 Toric blowing-up and a tower of toric blo""ing-ups 95 3 Dual Newton diagram and an admissible torie blowing-up 100 4 Resolution complexity . . . . . . . . . . 102 5 Characteristic power and Puiseux Pairs ...... 103 6 The Puiseux pairs of normal slice curves . . . . . . 106 7 Geometry of plane curves via a toroidal resolution 113 8 Iterated generic hyperplane section curves . . . . . 118
Introduction to the algorithm of resolution ORLANDO VILLAllilAYOR U.
1 Introduction ............ . 2 3 4 5 6
Stating the problem of resolution of singularities Auxiliary result: Idealistic pairs ......... . Constructive resolutions . . . . . . . . . . . . . . The language of groves and the problem of patching Examples ....................... .
II Complex Singularities and Differential Systems
Polarity with respect to a foliation J. GARciA A. J. REGUERA
1 Introduction ....... . 2 3 4 5 6
Preliminaries on linear systems The polarity map . PlUcker's formula The net of polars Some calculus ..
123 124 128 139 148 149
157 158 159 162 166 168
Table of Contents
On moduli spaces of semiquasihomogeneous singularities GERT-l\:IARTIN GREUEL GERHARD PFISTER
1 Introduction ................ . 2
3
4 5
Versal p-constant deformations and kernel of Kodaira-Spencer map . . . . . . . . . . . . . . . . . . . . Existence of a geometric quotient for fixed Hilbert function of -the Tjurina algebra . . . . . . . . . . . . . . . . . . . The automorphism group of semi Brieskorn singularities Problems ......................... .
Stratification Properties of Constructible Sets ZBIGNIEW HAJTO
1 Introduction........... 2 Grassmann blowing-up ..... 3 Analytically constructible sets . 4 An application: the Henry-1Ierle Proposition 5 Canonical stratification ........... .
On the linearization problem and some questions for webs in (:2
ALAIN HENAUT
1 Introduction in the form of a survey . . . . . . 2 Linearization of webs in (C2 , 0) . . . . . . . .. 3 Geometry of the abelian relation space and the
linearization problem in the maximum rank case 4 Some questions on webs in ([:2 . . . . • • . . . . .
Globalization of Admissible Deformations THEO DE JONG
1 Introduction ......... . 2 3
Compactification . . . . . . . Globalization of deformations
Caracterisation geometrique de l'existence du polynome de Bernstein relatif
J. BRIANQON PH. MAISO:-;OBE
vii
171
172
177 182 184
187 187 188 190 195
197 200
202 204
209 210 211
1 Polyn6me de Bernstein relatif . . . . . . . . . . . . . . . 216 2 V XxT Module holonome regulier relativement coherent 223
Le Polygone de Newton d'un Vx-module Z. MEBKHOUT
1 Introduction............ 237 2 Le cas d'une variable. . . . . . . 238 3 4
La categorie des faisceaux pervers . Le faisceau d'irregularite et Ie cycle d'irregularite
242 244
viii
5 6 7
Table of Contents
La filtration du faisceau d'irregularite ....... . Le polygone de Kewton d'un Vx-module ..... . Sur l'existence d'une equation fonctionnclle rcguliere
How good are real pictures?
DAVID MOND
1 Introduction..... 2 Comparison of real and complex discriminants and images 3 Codimension 1 germs. . . . . . . . . . . 4
5 Good real forms and their perturbations Bad real pictures . . . . . . . . . . . .
Weighted homogeneous complete intersections
C. T. C. WALL
1 Introduction: ..... . 2
3 4
5 6 7 8
Notation ........ . Ideals and C-equivalence Submodules . . . . . . . K-equivalence . . . . . . Combinatorial arguments A-equivalence . . . . Other ground fields . . . .
III Curves and Surfaces
Degree 8 and genus 5 curves in f>3 and the Horrocks-Mumford bundle.
M. R. GONZALEZ-DoRREGO
1 Construction of curves of degree 8 and genus 5 on a Kummer surface S E]p>3 • . • • . . . • . • •
2
3
Barth's Construction . . . . . . . . . . . . . . A generic curve of degree 8 and genus 5 in jp3
Irreducible Polynomials of k ( (X) ) [Y] A. GRANJA
247 250 253
259 262 264 266 271
277
278 281 283 285 291 294 298
303 305 306
1 Introduction.......... 311 ··2 Reduction of the Problem . . 312 3 Some Maximal Ideals of k[X~[Y] 313 4 Irreducibility Criterion for !>..fonic Polynomials of k[Xj[Y] 314 5 Some Ideas to Compute V[n/2](P) . . . . . . . . . . . . . . 316
Table of Contents
Examples of Abelian Surfaces with Polarization type (1,3) ISIDRO NIETO
1 Abstract ... 2 Introduction. 3 4 5
6 7 8
Preliminaries First examples: products of elliptic curves The two-dimensional families of T-invariant quartic surfaces ...... . The Family F.4.E . . . .
The Family t- 1 (Lo, 1, 2)
The Family F.4.B n F.4.E
Semigroups and Clusters at Infinity ANA-JOSE REGUERA LOPEZ
1 Introduction ........ . 2 3 4
The concept of approximant . . . Curves associated to a semigroup A family of examples . . . . . . .
Cubic surfaces with double points in positive characteristic MARKO ROCZEN
1 Introduction ................... . 2 Two characterizations of rational double points 3 Singularities and normal forms . . . . . . .
On the classification of reducible curve singularities JAN STEVENS
1 Reducible curve singularities. 2 Decomposable curves . . . . . 3 Classification......... 4 Deformations and smoothings
ix
319 319 320 321
326 329 334 336
339 341 347 360
375 375 378
384 385 392 401
Introduction
The volume contains both general and research papers. Among the first ones are papers showing recent and original developments or methods in subjects such as resolution of singularities, D-module theory, singularities of maps and geometry of curves. The research papers deal on topics related to, or close to, those listed_above. The contributions are organized in three parts according to their contents.
Part I presents a set of papers on resolution of singularities, a topic of renewed activity. It deals with important topics of current interest, such as canonical, algorithmic, combinatorial and graphical procedures (Villamayor, Oka, Marijmin), as well as special results on desingularization in characteristic p (Cossart, Moh), and connections between resolution and structure of the space of arcs through a singularity (Gonz81ez-Sprinberg-Lejeune-Jalabert).
Part II contains a series of papers on the study~of singularities and its connections with differential systems and deformation or perturbation theories. Two expository papers (Maisonobe-Briam;on, :'vlebkhout) describe, in an algebro-geometric way, the interaction between singularities and D-module t.heory including recent progress on Bernstein polynomials and Newton polygon techniques. Geometry of foliations (Henaut, Garcfa-Reguera), polar varieties and stratifications (Hajto) are also topics treated here. Two other papers (Wall, Greuel-Pfister) deal with quasihomogeneous singularities in the contexts of perturbations and moduli spaces. Globalization of deformations of singularities (de Jong) and determination of complex topology from the real one (~10nd) complete this series of papers.
Part III consists of papers on algebraic geometry of curves and surfaces. Equisingularity of plane curve singularities over arbitrary ground fields provides irreducibility criteria (Granja) and helps the study of singularities at infinity (Reguera). Classification of space curves is the objetive of another paper (Stevens). Curves on Kummer surfacp..s are not appropriate to construct the Horrocks- Mumford bundle (Gonzalez Dorrego). Two papers on surfaces with double points (Roczen) and abelian surfaces (Nieto) complete the volume.
The papers originated at the Third International Conference on Algebraic Geometry held at La Rabida (Spain) during the week of December 9-14, 1991. This time the main focus of the Conference was Singularity Theory, especially its algebraic aspects. This was an appropriate end of an active year in this field.
Several organizations sponsored the Conference: the European Singularity Project, the local government Junta de Andalucia, the cities of Palos de la Frontera and Huelva, and the universities of Valladolid and Sevilla. The Organizing Committee consisted of J.M. Aroca, T. Sanchez Giralda (Valladolid), F. Castro and J.L. Vicente (Sevilla). Profs. A. Campillo (Valladolid), G.M. Greuel (Kaiserslautern), Le Dung Trang (Paris), L Luengo (I\ladrid), L. Narvaez (Sevilla) formed the Scientific Committee.
xii Introduction
The Scientific Committee received a first version of the papers before November 30, 1992. Then the papers passed through a steady process of elaboration and refereeing. The final versions were produced before September 30, 1993. We thank the referees for their contributions. We also thank all speakers, participants, the Organizing and Scientific Committees, who helped, in one way or another, to make this book possible. We owe special thanks to J.L. Vicente for ma!dng possible again to enjoy the nice atmosphere of La Rabida. Antonio Ines organized the background tasks and i\:lanuel Soto did an excellent work by 1EXing the book.
A. CampiUo Lopez L. Narvaez Macarro
July, 1994
Plenary Conferences
BRASSELET, Jean Paul Le thCoreme de De Rham relatif
CASAS, Eduardo Singularities of polar curves
COSSART, Vincent Desingularization in dimension 3, characteristic p
LE DUNG TRANG Lefschetz theorem and vanishing of constructible cohomologies
LEJEUNE-JALABERT, Monique Sur l'espace des courbes des singularites de surface
MEBKHOUT, Zoghman Le polygne de Newton du faisceau d'irrgularit
MOH, T.T. On a Newton polygon approach to the uniformization of singularities of characteristic p
NAVARRO AZNAR, Vicente Connexion de Gauss-I\'1anin sur l'homotopie rationnelle
OKA, Mutsuo On the resolution complexity of planes curves and hyperplane sections
PHAM, Frederic Monodromy and resurgence
SAITO, Kyoji Jacobi inversion problem and root system
SIERSMA, Dirk Functions on singular spaces
SZPIRO, Lucien abc and Mordell
SPIVAKOSVSKI, Marc On the Artin approximation theorem
TEISS1ER, Bernard Local h-cobordism and the geometry of the real discriminant
WALL, Charles Terence Clagg Weighted homogeneous complete intersections
Specialized Conferences
ALEKSANDROV, Aleksandr Grigorjev;ch Deformations of zero-dimensional singularities
BALDASSARI, Francesco Generalized Hypergeometric Functions (GHF) and variation of cohomology (d'apres Dwork and Loeser)
BIRBRAIR, Gev The stratification of the space of homogeneous mappings
CASTELLANOS, Julio Arf closure relative to a divisorial valuation and transversal curves
DELGADO DE LA MATA, Felix Symmetry of semigroups and algebraic properties of curves
DU BOIS, Philippe Forme de Seifert des Entrelacs Algebriques
GAETA, Federico A discussion of the possible applications of the D-modules, hypergeometric functions, etc. to the Schottky problem
GARCIA DE LA FUENTE, Julio Polaridad relativa a una foliaci6n en p2
GONZALEZ DORREGO, Rosa Geometry and classification of Kummer surfaces in p3
GRANJA BARON, Angel Irreducible polynomials of k((X))[Y]
GREUEL, Gert-Martin A simple proof for the smoothness of the fl·-constant stratum for curves
HAJTO, Zbigniew Stratification Properties of Constructible Sets
HA. USER, Herwig and 1trULLER, Gert Auto~orphism groups in local analytic geometry, infinite dimensional rank theorem and Lie groups
HEN AUT, Alain Webs of maximun rank in ([:2 wieh are algebraic
xvi Specialized Conferences
HENRY, Jean-Pierre FRONCES et DOUBLES PLIS Towards a numC'rical criterion for Zariski equisingularity
JONG, Theo de Rational surface singularities ",ith reduced fundamental cycle and Globalization of admisible deformations
LANDO, Sergei Singularities of the differential forms of the highest degree and their deformations
LAURENT, Yves Irregular vanishing cycles for V-modules
MAISONOBE, Philippe Polyn6me de Bernstein relatif et type topologique constant
MARIJUAN LOPEZ, Carlos Desingularizaci6n geometrica de un grafo acfclico
MARTIN, Bernd Moduli spaces of singularities of simplest topological ty-pe
MERLE, Michel Multiplicites des varietes caracteristiques
MOND, David How good are real pictures?
MORALES, Marcel Blow-up of ideals of co dimension 2
MORENO SOCIAS, Guillermo An Ackermannian polynomial ideal
MULLER, Gerd Integral varieties of Lie algebras of vector fields
NIETO, Isidro Examples of Abelian surfaces with a level (2,6)-structure
PEREZ PEREZ, Tomas and FINAT CODES, Javier Lie Algebras preserving Tangent Spaces to R-orbits
PFISTER, Gerhard Moduli spaces of semiquasihomogeneous singularities
PISON CASARES, Pilar Monomial curves in A 4
Specialized Conferences
POLISCHUK, Alexander Noncommutative projective spaces
REGUERA LOPEZ, Ana Jose Plane curves associated to a semigroup
ROCZEN, Marko Recognition of simple singularities in positive characteristic
SABBAH, Claude Connexions mromorphes deux variables
STEIN, Harvey Singularities, Smooth Morphisms and Lifting Lemmas
STEVENS, Jan On the classification of reducible curve singularities
VILLAMAYOR, Orlando On constructive or algorithmic Resolution of singularities
XAMBO, Sebastin Rational equivalence on some families of plane curves
ZURRO MORO, M. Angeles Abhyankar-Jung revisited
x .... ii
List of Participants
Prof. ALEKSANDROV, Aleksandr Grigorjevich: Jvloscow, Urss. Prof. AROCA HERNANDEZ-ROS, J.M.; Valladolid, Espana. Prof. BALDASSARI, Francesco; Padova, Italia. Prof. BERMEJO DIAZ, :Marfa Isabel; Sta. Cruz de Tenerife, Espana. Prof. BIRBRAIR, Gev; Jerusalem, Israel. Prof. BRASSELET, Jean Paul; Luminy p.larsella), Francia. Prof. CABRERO VELASCO, Rafael; Valladolid, Espana. Prof. CAMPILLO L6PEZ, A.; Valladolid, Espana. Prof. CANO TORRES, Felipe; Valladolid, Espana. Prof. CANO TORRES, Jose; Valladolid, Espana. Prof. CARNICER, Manuel; Valladolid, Espana. Prof. CASAS ALVERO, E.; Barcelona, Espana. Prof. CASSOU-NOGUES, Pierrete; Talence, France. Prof. CASTELLANOS, Julio; Tenerife, Espana. Prof. CASTRO JIMENEZ, Francisco; Sevilla, Espana. Prof. CHARDIN, Marc; Palaiseau, France. Prof. COSSART, Vincent; Jouyen Josas, France. Prof. DELGADO DE LA MATA, Felix; Valladolid, Espana. Prof. DU BOIS, Philippe; Angers, France. Prof. ENCINAS CARRION, Santiago; Valladolid, Espana. Prof. FERNANDEZ DOMINGUEZ, Jesus; ??, 71. Prof. FERNANDEZ GUTIERREZ, Diego; Valladolid, Espana. Prof. FINAT CODES, Javier; Valladolid, Espana. Prof. GAETA, Federico; Madrid, Espana. Prof. GALINDO PASTOR, Carlos; Valladolid, Espana. Prof. GARCIA BARROSO, Evelia; Santa Cruz de Tenerife, Espana. Prof. GARCIA DE LA FUENTE, Julio; Valladolid, Espana. Prof. GIMENEZ, Philippe; Saint Martin d'Heres, France. Prof. GONZALEZ DORREGO, Rosa; Toronto, Canada. Prof. GONZALEZ SPRINBERG, Gerardo; Saint 1-fartin d'Heres, France. Prof. GRANGER, Jean Michel; Angers, France. Prof. GRANJA BARON, Angel; Leon, Espana. Prof. GREUEL, Gert-Martin; Kaiserlautern, Germany. Prof. GUDIEL RODRIGUEZ, Felix; Sevilla, Espana. Prof. GUEMES ALZAGA, Maria Belen; Valladolid, Espana. Prof. GUILLEN, F.; Barcelona, Espana. Prof. HAJTO, Zbigniew: Valladolid, Espana. Prof. IJAUSER, Herwig; Innsbruck, Austria. Prof. HENAUT, Alain; Talence, Francia. Prof. HENRY, Jean-Pierre; Palaiseau, France. Prof. HERNANDO IVIARTIN, :\iaria del Carmen; Barcelona, Espana. Prof. HIRONAKA, E.; Boon, Germany.
xx List of Participants
INES CALZON, Antonio; Sevilla, Espana. Prof. JONG, Theo de.; Kaiserlautern, Germany. Prof. KIYEK, Karl Heinz; Paderborn, Germany. Prof. LANDO, Sergei; Moscow, Urss. Prof. LAURENT, Yves; Saint :Martin d'Heres, France. Prof. LE DUNG TRANG; Paris, France. Prof. LEJEUNE-JALABERT, 1.fonique; Saint Martin d'Heres, France. Prof. LUENGO VELASCO, Ignacio; Madrid, Espana. Prof. MAHAMMED, Norreddine; Villeneuve d'Ascq., Francia. Prof. MAISONOBE, Philippe; Nice, France. Prof. MARIJUAN LOPEZ, Carlos; Valladolid, Espana. Prof. MARTIN, Bernd; Berlin, Germany. Prof. MARTINEZ MARTINEZ, Maria del Carmen; Valladolid. Espana. Prof. MEBKHOUT, Zoghman; Paris, Francia. Prof. MERLE, Michel; Kice, France. Prof. MOH, T.T.; Purdue, U.S.A. Prof. MOND, David; Coventry, Great Britain. Prof. MORALES, Marcel; Saint :1o,'lartin d'Heres, France. Prof. MORENO SOcIAS. Guillermo: Paris, France. Prof. MOZO FERKANDEZ, Jorge; . Valladolid, Espana. Prof. MULLER, Gerd; Mainz, Germany. Prof. NARVAEZ MACARRO, Luis; Sevilla, Espana. Prof. NATANZON, Sergej; Moscow, URSS. Prof. NAVARRO AZNAR, Vicente; Barcelona, Espana. Prof. NETO, Orlando; Lisboa, Portugal. Prof. NIETO, Isidro; 1·iexico D.F., Mexico. Prof. NUNEZ JIMENEZ, Carolina Ana; Valladolid, Espana. Prof. NUSS, Philippe; Strasbourg, France. Prof. OKA, Mutsuo; Tokyo, Japan. Prof. OLIVEIRA, Bruno; Lisboa, Portugal. Prof. PASCUAL GAINZA, Pere; Barcelona, Espana. Prof. PERAIRE DURBA, Rosa; Barcelona, Espana. Prof. PEREZ PEREZ, Tomas; Valladolid, Espana. Prof. PHAM, F.; Nice, France. Prof. PFISTER, Gerhard; Belin, Germany. Prof. PIEDRA SANCHEZ, Ramon: Sevilla, Espana. Prof. PISON CASARES, Pilar; Sevilla, Espana. Prof. POLISCHUK, Alexander; Moscow, URSS. Prof. POSISZELSKY, Leonid; :Moscou, URSS. Prof. REGUERA LOPEZ, Ana Jose; Valladolid, Espana. Prof. REY, Jerome; Toulouse, France. Prof. RIVERO ALVAREZ, Margarita; La Laguna (Tenerife), Espana. Prof. ROCZEN, Mark; Berlin, Germany. Prof. RODRIGUEZ SANCHEZ, Maria Cristina; Le6n, Espana.
List of Participants
Prof. SABBAH, Claude; Palaiseau, France. Prof. SAITO, K.; Kyoto, Japan. Prof. SANCHEZ GIRALDA, Tomas; Valladolid, Espana. Prof. SAULOY, Jacques; Toulouse, France. Prof. SERRANO, F.; Barcelona, Espana. Prof. SIERSMA, Dirk; Utrecht, Pays-Bas. Prof. SOLEEV, Ahmadjon; Samarkand, URSS. Prof. SPIVAKOSVSKI, M.; Toronto, Canada. Prof. STASICA,; Warsaw, Polland. Prof. STEIN, Harvey; Mexico D.F., Mexico. Prof. STEVENS, Jan; Hamburg, Germany. Prof. SZPIRGLAS, A viva; Letaneuse, France. Prof. SZPIRO, Lucien; Orsay, France. Prof. TEISSIER, Bernard; Paris, France. Prof. TIEP, Pham Huu; 1·1oscow, URSS. Prof. TRAN, Hoi Ngoc; Ho Chi 1.fihn City, Vietnam. Prof. VICENTE CORDOBA, Jose Luis; Sevilla, Espana. Prof. VILLA MAYOR, Orlando; Madrid, Espana. Prof. WALL, Charles Terence Clagg; Liverpool, Great Britain. Prof. XAMBO, S.; Madrid, Espana. Prof. XU AN HAl, Bui; Ho Chi ~1ihn City, Vietnam. Prof. ZURRO MORO, M. Angeles; Valladolid, Espana.
xxi