Test Ideals in Rings With Finitely Generated Anti-Canonical Algebras

Many results are known about test ideals and F -singularities for Q -Gorenstein rings. In this paper, we generalize many of these results to the case when the symbolic Rees algebra OX⊕ OX(-KX)⊕ OX(-2KX)⊕ ⋯ \, is finitely generated (or more generally, in the log setting for -KX-[STIX]x1D6E5 ). In particular, we show that the F -jumping numbers of [STIX]x1D70F(X,aᵗ) are discrete and rational. We show that test ideals [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.