TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS

Abstract

Many results are known about test ideals and$F$-singularities for$\mathbb{Q}$-Gorenstein rings. In this paper, we generalize many of these results to the case when the symbolic Rees algebra${\mathcal{O}}_{X}\oplus {\mathcal{O}}_{X}(-K_{X})\oplus {\mathcal{O}}_{X}(-2K_{X})\oplus \cdots \,$is finitely generated (or more generally, in the log setting for$-K_{X}-\unicode[STIX]{x1D6E5}$). In particular, we show that the$F$-jumping numbers of$\unicode[STIX]{x1D70F}(X,\mathfrak{a}^{t})$are discrete and rational. We show that test ideals$\unicode[STIX]{x1D70F}(X)$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.