Springer Monographs in Mathematics

Springer Monographs in Mathematics

G.-M. Greuel ? C. Lossen ? E. Shustin

Introduction to Singularities and Deformations

Gert-Martin Greuel Fachbereich Mathematik Universit?t Kaiserslautern Erwin-Schr?dinger-Str. 67663 Kaiserslautern, Germany e-mail: greuel@mathematik.uni-kl.de

Christoph Lossen Fachbereich Mathematik Universit?t Kaiserslautern Erwin-Schr?dinger-Str. 67663 Kaiserslautern, Germany e-mail: lossen@mathematik.uni-kl.de

Eugenii Shustin School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Ramat Aviv 69978 Tel Aviv, Israel e-mail: shustin@post.tau.ac.il

Library of Congress Control Number: 2006935374

Mathematics Subject Classification (2000): 14B05, 14B07, 14B10, 14B12, 14B25, 14Dxx, 14H15, 14H20, 14H50, 13Hxx, 14Qxx

ISSN 1439-7382 ISBN-10 3-540-28380-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-28380-5 Springer Berlin Heidelberg New York

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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Media c Springer-Verlag Berlin Heidelberg 2007

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Typesetting by the authors and VTEX using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg, Germany

Printed on acid-free paper SPIN: 10820313 VA 4141/3100/VTEX - 5 4 3 2 1 0

Meiner Mutter Irma und der Erinnerung meines Vaters Wilhelm

G.-M.G.

Fu?r Carmen, Katrin und Carolin C.L.

To my parents Isaac and Maya E.S.

VI

A deformation of a simple surface singularity of type E7 into four A1singularities. The family is defined by the equation

F (x, y, z; t) = z2 - x + 4t3 ? x2 - y2(y + t) . 27

The

pictures1

show

the

surface

obtained

for

t = 0,

t=

1 4

,

t=

1 2

and

t = 1.

1 The pictures were drawn by using the program surf which is distributed with Singular [GPS].

Preface

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, the theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. The specific feature of the present Introduction to Singularities and Deformations, separating it from other introductions to singularity theory, is the choice of a material and a unified point of view based on the theory of analytic spaces.

This text has grown up from a preparatory part of our monograph Singular algebraic curves (to appear), devoted to the up-to-date theory of equisingular families of algebraic curves and related topics such as local and global deformation theory, the cohomology vanishing theory for ideal sheaves of zerodimensional schemes associated with singularities, applications and computational aspects. When working at the monograph, we realized that in order to keep the required level of completeness, accuracy, and readability, we have to provide a relevant and exhaustive introduction. Indeed, many needed statements and definitions have been spread through numerous sources, sometimes presented in a too short or incomplete form, and often in a rather different setting. This, finally, has led us to the decision to write a separate volume, presenting a self-contained textbook on the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities.

Having in mind to get the reader ready for understanding the volume Singular algebraic curves, we did not restrict the book to that specific purpose. The present book comprises material which can partly be found in other books and partly in research articles, and which for the first time is exposed from a unified point of view, with complete proofs which are new in many cases. We include many examples and exercises which complement and illustrate the general theory. This exposition can serve as a source for special courses in singularity theory and local algebraic and analytic geometry. A special attention is paid to the computational aspect of the theory, illustrated by a

VIII Preface

number of examples of computing various characteristics via the computer algebra system Singular [GPS]2. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.

In the first part of the book we develop the relevant techniques, the basic theory of complex spaces and their germs and sheaves on them, including the key ingredients - the Weierstra? preparation theorem and its other forms (division theorem and finiteness theorem), and the finite coherence theorem. Then we pass to the main object of study, isolated hypersurface and plane curve singularities. Isolated hypersurface singularities and especially plane curve singularities form a classical research area which still is in the centre of current research. In many aspects they are simpler than general singularities, but on the other hand they are much richer in ideas, applications, and links to other branches of mathematics. Furthermore, they provide an ideal introduction to the general singularity theory. Particularly, we treat in detail the classical topological and analytic invariants, finite determinacy, resolution of singularities, and classification of simple singularities.

In the second chapter, we systematically present the local deformation theory of complex space germs with an emphasis on the issues of versality, infinitesimal deformations and obstructions. The chapter culminates in the treatment of equisingular deformations of plane curve singularities. This is a new treatment, based on the theory of deformations of the parametrization developed here with a complete treatment of infinitesimal deformations and obstructions for several related functors. We further provide a full disquisition on equinormalizable (-constant) deformations and prove that after base change, by normalizing the -constant stratum, we obtain the semiuniversal deformation of the parametrization. Equisingularity is first introduced for deformations of the parametrization and it is shown that this is essentially a linear theory and, thus, the corresponding semiuniversal deformation has a smooth base. By proving that the functor of equisingular deformations of the parametrization is isomorphic to the functor of equisingular deformations of the equation, we substantially enhance the original work by J. Wahl [Wah], and, in particular, give a new proof of the smoothness of the -constant stratum. Actually, this part of the book is intended for a more advanced reader familiar with the basics of modern algebraic geometry and commutative algebra. A number of illustrating examples and exercises should make the material more accessible and keep the textbook style, suitable for special courses on the subject.

Cross references to theorems, propositions, etc., within the same chapter are given by, e.g., "Theorem 1.1". References to statements in another chapter are preceded by the chapter number, e.g., "Theorem I.1.1".

2 See [GrP, DeL] for a thorough introduction to Singular and its applicability to problems in algebraic geometry and singularity theory.

Preface IX

Acknowledgements

Our work at the monograph has been supported by the Herman Minkowsky? Minerva Center for Geometry at Tel?Aviv University, by grant no. G-61615.6/99 from the German-Israeli Foundation for Research and Development and by the Schwerpunkt "Globale Methoden in der komplexen Geometrie" of the Deutsche Forschungsgemeinschaft. We have significantly advanced in our project during our two "Research-in-Pairs" stays at the Mathematisches Forschungsinstitut Oberwolfach. E. Shustin was also supported by the Bessel Research Award from the Alexander von Humboldt Foundation.

Our communication with Antonio Campillo, Steve Kleiman and Jonathan Wahl was invaluable for successfully completing our work. We also are very grateful to Thomas Markwig, Ilya Tyomkin and Eric Westenberger for proofreading and for partly typing the manuscript.

Kaiserslautern - Tel Aviv, August 2006

G.-M. Greuel, C. Lossen, and E. Shustin

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