Test ideals in rings with finitely generated anti-canonical algebras

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 $O_X \oplus O_X(-K_X) \oplus O_X(-2K_X) \oplus …$isfinitelygenerated(ormoregenerally,inthelogsettingfor$-K_X - \Delta$)\mathrm{.}Inparticular,weshowthatthe$F$-jumpingnumbersof$\tau(X, a^t)$arediscreteandrational\mathrm{.}Weshowthattestideals$\tau(X)$ 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.