Singularities of determinantal pure pairs

Determinantal singularities and Newton polyhedra

We introduce so called resultantal singularities (Definitions 5.1 and 8.1), whose study in terms of Newton polyhedra unifies the tasks A and B to a certain extent. In particular, this provides new formulations and proofs for a number of well known results (see, for example, Theorem 5.7 related to task A and Corollary 4.6 related to task B). As an application, we study basic topological invarian...

Singularities of Pairs

Higher dimensional algebraic geometry has been one of the most rapidly developing research areas in the past twenty years. The first decade of its development centered around the formulation of the minimal model program and finding techniques to carry this program through. The proof of the existence of flips, given in [Mori88], completed the program in dimension three. These results, especially...

Rational Singularities Associated to Pairs

Rational singularities are a class of singularities which have been heavily studied since their introduction in the 1960s. Roughly speaking, an algebraic variety has rational singularities if its structure sheaf has the same cohomology as the structure sheaf of a resolution of singularities. Rational singularities enjoy many useful properties, in particular they are both normal and Cohen-Macaul...

Rings of Singularities

This paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``Advanced School and Conference on homological and geometrical methods in representation theory'' at the International Centre for Theoretical Physics (ICTP), Miramare - Trieste, Italy, 11-29 January 2010. We show how to associate to a triple of posit...

The multiplicity of pairs of modules and hypersurface singularities