Introduction to Singularities and Deformations

  • Gert-Martin Greuel
  • Christoph Lossen
  • Eugenii Shustin

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-XII
  2. Pages 1-220
  3. Pages 221-377
  4. Back Matter
    Pages 379-474

About this book

Introduction

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 with numerous applications in the natural and technical sciences.

This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete.

In the first part of the book the authors develop the relevant techniques, including the Weierstraß preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.

The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thus can serve as source for special courses in singularity theory and local algebraic and analytic geometry.

Keywords

Algebra Complex space germs Equisingularity Local and semi-local rings Plane curve singularities Versal deformation deformation theory singularity theory

Authors and affiliations

  • Gert-Martin Greuel
    • 1
  • Christoph Lossen
    • 2
  • Eugenii Shustin
    • 3
  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  3. 3.School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/3-540-28419-2
  • Copyright Information Springer-Verlag Berlin Heidelberg 2007
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-28380-5
  • Online ISBN 978-3-540-28419-2
  • Series Print ISSN 1439-7382
  • About this book