Introduction to Singularities and Deformations
BeschreibungThis 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.
InhaltsverzeichnisI. 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
PressestimmenFrom the reviews: "This monograph is dedicated to the theory of singularities, a subject with a central role in modern mathematics. ... This very well written book has a unified point of view based on the theory of analytic spaces, which allows a coherent presentation of both of its main themes: the theory of singularities and deformations of singularities. ... The book includes many examples and exercises ... . This monograph can serve as a source for several special courses in singularity theory and local analytic geometry." (Vasile Brînzanescu, Mathematical Reviews, Issue 2008 b)
Untertitel: 'Springer Monographs in Mathematics'. 2007. Auflage. 54 schwarz-weiße Abbildungen, Bibliographie. Book. Sprache: Englisch.
Verlag: Springer-Verlag GmbH
Erscheinungsdatum: November 2006