F-thresholds, integral closure and convexity

TL;DR

This paper explores the relationship between F-thresholds, integral closure, and convexity, proposing a new perspective on computing F-jumping numbers using a special polynomial under certain convexity assumptions.

Contribution

It introduces a novel approach linking convexity patterns of integral closures to the computation of F-jumping numbers via a specific polynomial.

Findings

If integral closures follow convexity patterns, a polynomial exists to compute F-jumping numbers.

The paper suggests potential sources of examples fitting this framework.

Provides a new perspective on the interplay between algebraic properties and convexity.

Abstract

The purpose of this note is to revisit the results of arXiv:1407.4324 from a slightly different perspective, outlining how, if the integral closures of a finite set of prime ideals abide the expected convexity patterns, then the existence of a peculiar polynomial allows to compute the F-jumping numbers of all the ideals formed by taking sums of products of the original ones. The note concludes with the suggestion of a possible source of examples falling in such a framework.