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TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 13 January 2016, pp. 171-206
- Print publication:
- February 2018
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- Article
- Export citation
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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.
