Jumping numbers and ordered tree structures on the dual graph

TL;DR

This paper explores the jumping numbers of complete ideals in two-dimensional regular local rings using dual graph structures, providing a combinatorial criterion for identifying these numbers.

Contribution

It introduces a novel combinatorial criterion for jumping numbers based on ordered tree structures on the dual graph of minimal log resolutions.

Findings

Provides a combinatorial criterion for jumping numbers.

Associates ordered tree structures to each jumping number.

Enhances understanding of the relationship between dual graphs and jumping numbers.

Abstract

Let R be a two-dimensional regular local ring having an algebraically closed residue field and let a be a complete ideal of finite colength in R. In this article we investigate the jumping numbers of a by means of the dual graph of the minimal log resolution of the pair (X,a). Our main result is a combinatorial criterium for a positive rational number to be a jumping number. In particular, we associate to each jumping number certain ordered tree structures on the dual graph.