Poincar\'e series of multiplier ideals in two-dimensional local rings with rational singularities

Abstract

We study the multiplicity of the jumping numbers of an $\mathfrak m$-primary ideal $\mathfrak a$ in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincar\'e series of multiplier ideals associated to $\mathfrak a$ proving, in particular, that it is a rational function.