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

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.