Abstract
A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly |$F$|-regular ring is Gorenstein, in terms of an |$F$|-pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal |$F$|-pure (respectively log canonical) singularity is quasi-Gorenstein, in terms of an |$F$|-pure (respectively log canonical) threshold.
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