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 the 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, and 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.

It is a fairly self-contained introduction to Singularity Theory and to Deformation Theory in any characteristic Provides a comprehensible and useful source for graduate students, researchers, lecturers, and practitioners Provides many algorithms and code that can be executed with the computer algebra system SINGULAR

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), 19801981 Professor at Univ. Osnabrück (C3), 19812010 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

1 Singularity Theory.- 2 Local Deformation Theory.- 3 Singularities in Arbitrary Characteristics.- Appendix A: Sheaves.- Appendix B: Commutative Algebra.- Appendix C: Formal Deformation Theory.