Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences.
Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory and the theory of plane curve singularities. It includes complete proofs.
From 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 Brinzanescu, Mathematical Reviews, Issue 2008 b)
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