Test ideals in rings with finitely generated anti-canonical algebras

Abstract

Many results are known about test ideals and $F$-singularities for $\mathbf{Q}$-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.

Lemma 2.10 was incorrect, it stated something that was too strong. We have restated Lemma 2.10 correctly we believe. Fortunately, we only needed the new weaker statement. This also corrects the published version

References in corpus (15)

• Globally $F$-regular and log Fano varieties
• Singularities on normal varieties
• On the behavior of test ideals under finite morphisms
• Discreteness and rationality of $F$-jumping numbers on singular varieties
• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
• Q-Gorenstein splinter rings of characteristic p are F-regular
• Derived splinters in positive characteristic
• Test ideals via algebras of $p^{-e}$-linear maps
• The non-nef locus in positive characteristic
• Rings of Frobenius operators
• Supplements to non-lc ideal sheaves
• On base point free theorem of threefolds in positive characteristic
• On base point freeness in positive characteristic
• Comparing multiplier ideals to test ideals on numerically Q-Gorenstein varieties
• About a Minimal Model Program without flips

Cited by in corpus (4)

• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• A Gorenstein criterion for strongly $F$-regular and log terminal singularities
• $D$-modules, Bernstein-Sato polynomials and $F$-invariants of direct summands
• The vanishing conjecture for maps of Tor and derived splinters