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

TL;DR

This paper investigates the structure of multiplier ideals in two-dimensional local rings with rational singularities, providing formulas for jumping number multiplicities and demonstrating that their Poincaré series is a rational function.

Contribution

It introduces a simple method to detect jumping numbers and explicitly describes the Poincaré series of multiplier ideals in this setting.

Findings

Multiplicities of jumping numbers can be computed efficiently.

The Poincaré series of multiplier ideals is a rational function.

A new explicit description of the Poincaré series is provided.

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.