Introduction to Singularities and Deformations

This book presents the basic singularity theory of analytic spaces and the local deformation theory of plane curve singularities. In the first part the authors develop the relevant techniques, including Weierstraß preparation theorem, the finite coherence theorem etc., and then locally treat 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 put on the issues of the versality, obstructions, equisingular deformations. The book contains moreover a full and 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 parametrization. The material, which can partly be found in other books and partly in research articles, is for the first time exposed from a unified point of view, is supplied by complete proofs (new in many cases), and can serve as source for special courses in singularity theory. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from the sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained.