Introduction to Singularities and Deformations

This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. The authors develop the relevant techniques, including Weierstraß preparation theorem, the finite coherence theorem etc., and then discuss isolated hypersurface and plane curve singularities, including the finite determinacy, classification of simple singularities, topological and analytic invariants, resolution. In the local deformation theory emphasis is placed on the issues of the versality, obstructions, and equisingular deformations. The book includes a thorough treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum based on deformations of the parametrization.

The material is for the first time exposed from a unified point of view Material is supplied with complete proofs (new in many cases), and can serve as source for special courses in singularity theory. Three appendices, including basic facts from the sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained Includes supplementary material: sn.pub/extras

Autorentext

Gert-Martin Greuel: Born 1944, Studies of Mathematics and Physics at Univ. Göttingen and ETH Zürich, Diploma 1971, PhD 1973 (Göttingen) and Habilitation 1980 in Mathematics (Bonn), 1980–1981 Professor at Univ. Osnabrück (C3), 1981–2010 Professor Univ. Kaiserslautern (C4), 2010 – 2015 Distinguished Senior Professor at Univ. Kaiserslautern, 2015 Emeritus. 2002 – 2013 Director of Mathematisches Forschungsinstitut Oberwolfach, 2009 Dr.h.c. from Leibniz Univ. Hannover, 2011 Honorary Member of Real Sociedad Matemática Española, 2012 – 2015 Editor-in-Chief of Zentralblatt MATH (zbMATH), 2004 First Richard D. Jenks Prize for Excellence in Software Engineering to the Singular team, 2013 Media Prize Mathematik by Deutsche Mathematiker Vereinigung. Christoph Lossen: Born in 1967, Study of mathematics and economical sciences at the University of Kaiserslautern, 1994 Diploma in Mathematics, 1998 PhD at the University of Kaiserslautern, 2002 State doctorate (Habilitation), 2002-2006 Assistant Professor (Hochschuldozent) at TU Kaiserslautern, since 2006 Administrative Director of the Department of Mathematics at TU Kaiserslautern (since 2023 University of Kaiserslautern-Landau (RPTU)). Eugenii I. Shustin: Born 1957, Studies of Mathematics at Leningrad State Univ. and Gorky State Univ., Ms. 1979, PhD 1984 (Leningrad). 1984-87 Assistant Prof. at Gorky Civil Eng. Inst., 1987-92 Associate Prof. Kuibyshev State Univ., 1992-96 Associate Prof. Tel Aviv Univ., 1996-now Full Prof. Tel Aviv University. 1990 Invited lecturer at ICM[1]90, Kyoto, 2002 Bessel Research Award from Alexander von Humboldt Foundation, 2018-now The Bauer-Neuman Chair in Real and Complex Geometry.

Inhalt

I. Singularity Theory. Basic Properties of Complex Spaces and Germs. Weierstrass Preparation and Finiteness Theorem. Application to Analytic Algebras. Complex Spaces. Complex Space Germs and Singularities. Finite Morphisms and Finite Coherence Theorem. Applications of the Finite Coherence Theorem. Finite Morphisms and Flatness. Flat Morphisms and Fibres. Singular Locus and Differential Forms. Hypersurface Singularities. Invariants of Hypersurface Singularities. Finite Determinacy. Algebraic Group Actions. Classification of Simple Singularities. Plane Curve Singularities. Parametrization. Intersection Multiplicity. Resolution of Plane Curve Singularities. Classical Topological and Analytic Invariants.- II. Local Deformation Theory. Deformations of Complex Space Germs. Deformations of Singularities. Embedded Deformations. Versal Deformations. Infinitesimal Deformations. Obstructions. Equisingular Deformations of Plane Curve Singularities.- Equisingular Deformations of the Equation. The Equisingularity Ideal. Deformations of the Parametrization. Computation of T^1 and T^2 . Equisingular Deformations of the Parametrization. Equinormalizable Deformations. Versal Equisingular Deformations.- Appendices: Sheaves. Commutative Algebra. Formal Deformation Theory. Literature.- Index.