Discreteness and rationality of $F$-jumping numbers on singular varieties

TL;DR

This paper proves that the $F$-jumping numbers of test ideals on certain singular varieties are always discrete and rational, extending known results from characteristic zero to positive characteristic settings.

Contribution

It establishes the discreteness and rationality of $F$-jumping numbers under broad conditions, generalizing previous results in the theory of singularities in positive characteristic.

Findings

$F$-jumping numbers are discrete and rational under specified conditions.

Results extend known characteristic zero properties to positive characteristic.

Applicable to a wide class of singular varieties with certain conditions.

Abstract

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.