Ascending chain condition for $F$-pure thresholds with fixed embedding dimension

TL;DR

This paper proves that the set of all $F$-pure thresholds for ideals with fixed embedding dimension satisfies the ascending chain condition, and verifies this for $d$-dimensional normal l.c.i. varieties, also establishing rationality in certain cases.

Contribution

It establishes the ascending chain condition for $F$-pure thresholds with fixed embedding dimension and for $d$-dimensional normal l.c.i. varieties, and proves rationality in specific contexts.

Findings

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

Verified ascending chain condition for $F$-pure thresholds on $d$-dimensional normal l.c.i. varieties.

Proved rationality of $F$-pure thresholds on non-strongly $F$-regular pairs.

Abstract

In this paper, we prove that the set of all $F$-pure thresholds of ideals with fixed embedding dimension satisfies the ascending chain condition. As a corollary, given an integer $d$, we verify the ascending chain condition for the set of all $F$-pure thresholds on all $d$-dimensional normal l.c.i. varieties. In the process of proving these results, we also show the rationality of $F$-pure thresholds of ideals on non-strongly $F$-regular pairs.