Singularities in Algebraic and Analytic Geometry

Topics include the role of valuation theory in algebraic geometry with recent applications to the structure of morphisms; algorithmic approaches to resolution of equisingular surface singularities and locally toric varieties; weak subintegral closures of ideals and Rees valuations; constructions of universal weakly subintegral extensions of rings; direct-sum decompositions of finitely generated modules; construction and examples of resolution graphs of surface singularities; Jacobians of meromorphic curves; investigation of spectral numbers of curve singularities using Puiseux pairs; Gröbner basis calculations of Hochschild homology for hypersurfaces with isolated singularities; and the theory of characteristic classes of singular spaces--a brief history with conjectures and open problems.

Contents

• S. S. Abhyankar and A. Assi -- Factoring the Jacobian
• M. A. Vitulli -- Weak normalization and weak subintegral closure
• L. G. Roberts -- Integral dependence and weak subintegrality
• R. Wiegand -- Singularities and direct-sum decompositions
• S. D. Cutkosky -- Valuations in algebra and geometry
• C. Ban and L. J. McEwan -- Simultaneous resolution of equisingular quasi-ordinary singularities
• C. G. Melles and P. Milman -- Single-step combinatorial resolution via coherent sheaves of ideals
• A. Némethi -- Resolution graphs of some surface singularities, I. (Cyclic coverings)
• A. Némethi and Szilárd -- Resolution graphs of some surface singularities, II. (Generalized Iomdin series)
• L. J. McEwan -- Inequalities for spectral distributions of curve singularities
• R. I. Michler -- Isolated singularities with large Hochschild homology
• J.-P. Brasselet -- Milnor classes via polar varieties