Discreteness of $F$-jumping numbers at isolated non-Q-Gorenstein points

TL;DR

This paper proves that $F$-jumping numbers are discrete at isolated non-Q-Gorenstein points under certain conditions and extends the results to characteristic zero, providing insights into the structure of singularities.

Contribution

It establishes the discreteness of $F$-jumping numbers at isolated points when the symbolic Rees algebra is finitely generated outside those points and generalizes related stabilization theorems.

Findings

$F$-jumping numbers have no limit points under specified conditions

Discreteness holds when the symbolic Rees algebra is finitely generated outside isolated points

Generalization of the Hartshorne-Speiser-Lyubeznik-Gabber stabilization theorem

Abstract

We show that the $F$-jumping numbers of a pair $(X, \mathfrak a)$ in positive characteristic have no limit points whenever the symbolic Rees algebra of $-K_X$ is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne-Speiser-Lyubeznik-Gabber stabilization theorem describing the non-$F$-pure locus of a variety.