A note on discreteness of $F$-jumping numbers

TL;DR

This paper proves the discreteness of $F$-jumping numbers in certain algebraic settings, showing they have no limit points under specific conditions without restrictions on the $Q$-Gorenstein index.

Contribution

It establishes the discreteness of $F$-jumping numbers for test ideals in normal, $Q$-Gorenstein rings without restrictions on the index, extending previous results.

Findings

$F$-jumping numbers have no limit points in normal, $Q$-Gorenstein rings.

Discreteness of $F$-jumping numbers holds when $K_R + riangle$ is $R$-Cartier.

Results do not require the $Q$-Gorenstein index to be coprime with $p$.

Abstract

Suppose that $R$ is a ring essentially of finite type over a perfect field of characteristic $p > 0$ and that $a \subseteq R$ is an ideal. We prove that the set of $F$-jumping numbers of $\tau_b(R; a^t)$ has no limit points under the assumption that $R$ is normal and $Q$-Gorenstein -- we do \emph{not} assume that the $Q$-Gorenstein index is not divisible by $p$. Furthermore, we also show that the $F$-jumping numbers of $\tau_b(R; \Delta, a^t)$ are discrete under the more general assumption that $K_R + \Delta$ is $\bR$-Cartier.