Ascending chain condition for $F$-pure thresholds on a fixed strongly $F$-regular germ

TL;DR

This paper proves that the set of all $F$-pure thresholds on a fixed strongly $F$-regular germ satisfies the ascending chain condition, confirming a conjecture for certain varieties.

Contribution

It establishes the ascending chain condition for $F$-pure thresholds on a fixed germ, extending to smooth and tame quotient singularities, and confirms a conjecture by Blickle, Mustață, and Smith.

Findings

Set of $F$-pure thresholds satisfies ascending chain condition

Confirmed conjecture for smooth varieties and tame quotient singularities

Provides a foundational result for $F$-singularity theory

Abstract

In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Musta\c{t}\v{a} and Smith.