Jumping Numbers on Algebraic Surfaces with Rational Singularities

TL;DR

This paper develops an algorithm to compute jumping numbers of ideals at rational singularities on algebraic surfaces, linking resolution data to these invariants and applying the method to plane curves and specific surface singularities.

Contribution

It introduces a novel algorithm for calculating jumping numbers using resolution divisors, applicable to complex surface singularities and plane curves.

Findings

Algorithm for computing jumping numbers from resolution data

Explicit calculations for Du Val and toric surface singularities

Application to smooth surface cases

Abstract

In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.