Multiplicity of filtered rings and simple $K3$ singularities of multiplicity two

The Multiplicity Polar Theorem and Isolated Singularities

Recently there has been much work done in the study of isolated singularities. The singularities may be those of sets, functions, vector fields or differential forms. Examples can be found in the references at the end of this paper. In these works the authors have developed algebraic invariants which describe the isolated singularity. In this paper, we show how many of these invariants can be c...

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 Equality I = Qi in Buchsbaum Rings with Multiplicity Two

Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I = Q : m. Then the equality I = QI holds true, if e(A) = 2 and depth A > 0. The assertion is no longer true, unless e(A) = 2. Counterexamples are given.

Rings That Are Homologically of Minimal Multiplicity

Let R be a local Cohen-Macaulay ring with canonical module ωR. We investigate the following question of Huneke: If the sequence of Betti numbers {β i (ωR)} has polynomial growth, must R be Gorenstein? This question is well-understood when R has minimal multiplicity. We investigate this question for a more general class of rings which we say are homologically of minimal multiplicity. We provide ...

The multiplicity of pairs of modules and hypersurface singularities