Multiplicity of Filtered Rings and Simple K3 Singularities of Multiplicity Two

Abstract

Given a filtered ring, we give bounds of its multiplicity in terms of the data of the tangent cone using the technique of the filtered blowing-up. Applying it to each simple K3 singularity of multiplicity two, we find a good coordinate where the Newton boundary of the defining equation contains the point $(1,1,1,1)\in {\mathbf{R}}^4$. In the course of the proof, we classify simple K3 singularities of multiplicity two into 48 weight types. Furthermore we prove that the weight type of the singularity stays the same under arbitrary one-parameter (FG)-deformations.