Poincaré series of curves on rational surface singularities

Abstract

For a reducible curve singularity embedded in a rational surface singularity the Poincaré series is computed. Here the Poincaré series is defined by the multi-index filtration on the local ring defined by orders of a function on the branches of the curve. The method of the computations is based on the notion of the integral with respect to the Euler characteristic over the projectivization of the ring of functions (notion similar to, and inspired by, the notion of motivic integration). For the case of the E_8 surface singularity it appears that the Poincaré series coincides with the Alexander polynomial of the corresponding link.