Deformations of quasi-homogeneous surface singularities

Abstract

Let A = ~ A~ be a finitely generated normal graded domain over (E, so that i=0 V= Spec A has an isolated singularity. We say V is a quasi-homogeneous singularity, or admits a good IE*-action, or is defined by weighted homogeneous polynomials. We are concerned with two questions about the deformations of V. There are two natural notions of "equisingular deformation" of V. First, one has deformations of weight > 0; that is, one perturbs the defining equations of Vby terms of weight at least as big. To make a good notion out of this, Dock Sang Rim observed one is really deforming A and the weight filtration {Ij}, where Ij = ~ Ai ;