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arXiv · 1412.6453

Test ideals in rings with finitely generated anti-canonical algebras

Alberto Chiecchio·Florian Enescu·Lance Edward Miller·Karl Schwede
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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.

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Alberto Chiecchio, Florian Enescu, Lance Edward Miller, Karl Schwede. 2014-12-19. Test ideals in rings with finitely generated anti-canonical algebras. https://doi.org/10.1017/s1474748015000456

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