A Gorenstein criterion for strongly $F$-regular and log terminal singularities

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 (resp. log canonical) singularity is quasi-Gorenstein, in terms of an $F$-pure (resp. log canonical) threshold.

arXiv commentThis version corrects some errors in Proposition 4.9, Remark 4.10, and Proposition 4.13; the main theorems stay the same

References in corpus (8)

• Globally $F$-regular and log Fano varieties
• Inversion of adjunction on log canonicity
• Singularities on normal varieties
• The canonical sheaf of Du Bois singularities
• F-singularities of pairs and Inversion of Adjunction of arbitrary codimension
• Test ideals in rings with finitely generated anti-canonical algebras
• F-thresholds of graded rings
• Nilpotence of Frobenius action and the Hodge filtration on local cohomology