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 − ∆). In particular, we show that the F -jumping numbers of τ (X, a t ) are discrete and rational. We show that test ideals τ (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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