F-pure thresholds of homogeneous polynomials

TL;DR

This paper studies F-pure thresholds of homogeneous polynomials with isolated singularities, linking them to log canonical thresholds via base p expansion, and explores bounds, prime sets, and conjectures related to these invariants.

Contribution

It characterizes F-pure thresholds for homogeneous polynomials in terms of log canonical thresholds and base p expansion, advancing understanding of their relationship and properties.

Findings

Characterization of F-pure thresholds via base p expansion.

Bounds on the difference between F-pure and log canonical thresholds.

Results on the ACC conjecture and semi-continuity of F-pure thresholds.

Abstract

In this article, we investigate F-pure thresholds of polynomials that are homogeneous under some N-grading, and have an isolated singularity at the origin. We characterize these invariants in terms of the base p expansion of the corresponding log canonical threshold. As an application, we are able to make precise some bounds on the difference between F-pure and log canonical thresholds established by Musta\c{t}\u{a} and the fourth author. We also examine the set of primes for which the F-pure and log canonical threshold of a polynomial must differ. Moreover, we obtain results in special cases on the ACC conjecture for F-pure thresholds, and on the upper semi-continuity property for the F-pure threshold function.