On the arc filtration for the singularities of Arnold's lists

Abstract

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincar\'e series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincar\'e series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. Besides that we give a general formula for the Poincar\'e series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.