A formula for jumping numbers in a two-dimensional regular local ring

TL;DR

This paper provides an explicit formula for calculating jumping numbers of finite colength ideals in two-dimensional regular local rings, using numerical semigroups associated with the dual graph of a log-resolution.

Contribution

It introduces a novel explicit formula linking jumping numbers to numerical semigroups derived from the dual graph of a log-resolution.

Findings

Explicit formula for jumping numbers in 2D regular local rings

Association of numerical semigroups to dual graph vertices

Application to ideals with finite colength

Abstract

In this article we give an explicit formula for the jumping numbers of an ideal of finite colenght in a two-dimensional regular local ring with an algebraically closed residue field. For this purpose, we associate a certain numerical semigroup to each vertex of the dual graph of a log-resolution of the ideal.