Multiplier ideals in two-dimensional local rings with rational singularities

TL;DR

This paper investigates the structure of multiplier ideals and jumping numbers in two-dimensional local rings with rational singularities, providing an algorithm for their computation and introducing the concept of jumping divisors.

Contribution

It introduces a method to determine the next jumping number from multiplier ideals and develops the notion of jumping divisors, including a minimal jumping divisor, for detailed analysis.

Findings

Established an algorithm for computing jumping numbers and multiplier ideals.

Defined and studied the minimal jumping divisor.

Connected multiplier ideals with the geometry of rational singularities.

Abstract

The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is studied extensively.