Test ideals in rings with finitely generated anti-canonical algebras

TL;DR

This paper extends the theory of test ideals and $F$-singularities to rings with finitely generated anti-canonical algebras, establishing key properties and connections to multiplier ideals in this broader context.

Contribution

It generalizes known results about test ideals to rings with finitely generated anti-canonical algebras, including discreteness of $F$-jumping numbers and descriptions via alterations.

Findings

$F$-jumping numbers are discrete and rational.

Test ideals can be described by alterations, implying splinters are strongly $F$-regular.

Multiplier ideals reduce to test ideals under reduction modulo $p$.

Abstract

Many results are known about test ideals and $F$-singularities for ${\bf Q}$-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus ...$ is finitely generated (or more generally, in the log setting for $-K_X - \Delta$). In particular, we show that the $F$-jumping numbers of $\tau(X, a^t)$ are discrete and rational. We show that test ideals $\tau(X)$ can be described by alterations as in Blickle-Schwede-Tucker (and hence show that splinters are strongly $F$-regular in this setting -- recovering a result of Singh). We demonstrate that multiplier ideals reduce to test ideals under reduction modulo $p$ when the symbolic Rees algebra is finitely generated. We prove that Hartshorne-Speiser-Lyubeznik-Gabber type stabilization still holds. We also show that test ideals satisfy global generation properties in this setting.