Introduction to Singularities and Deformations
Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and wi
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G.-M. Greuel • C. Lossen • E. Shustin
Introduction to Singularities and Deformations
Gert-Martin Greuel Fachbereich Mathematik Universität Kaiserslautern Erwin-Schrödinger-Str. 67663 Kaiserslautern, Germany e-mail: [email protected] Christoph Lossen Fachbereich Mathematik Universität Kaiserslautern Erwin-Schrödinger-Str. 67663 Kaiserslautern, Germany e-mail: [email protected] 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: [email protected] 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 springer.com 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. F¨ ur 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 A1 singularities. The family is defined by the equation 4t3 · x2 − y 2 (y + t) . F (x, y, z; t) = z 2 − x + 27 The pictures1 show the surface obtained for t = 0, t = 14 , t =
1
1 2
and t = 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 materi
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