A Gorenstein Criterion for StronglyF-Regular and Log Terminal Singularities

Abstract

Dedicated to Professor Craig Huneke on the occasion of his sixty-fifth birthday.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.

Hankel determinantal rings have rational singularities

“…The definition of F-pure thresholds is due to Takagi and Watanabe [TW], and provides a positive characteristic analogue of the log canonical threshold. We focus here on the F-pure threshold of a homogeneous ideal in standard graded F-pure ring: Suppose, in addition, that R is normal; let ω R be the graded canonical module of R. Taking a to be m R in the above definition, [STV,Theorem 4.1] implies that −ν e (m R ) equals the degree of a minimal generator of ω (1−q) R . Using this, we obtain:…”

“…Let a be a homogeneous ideal of R, and let J be its preimage in A. Given e ∈ N, set ν e (a) := max r 0 | (I [q] : A I)J r ⊆ m Suppose, in addition, that R is normal; let ω R be the graded canonical module of R. Taking a to be m R in the above definition, [STV,Theorem 4.1] implies that −ν e (m R ) equals the degree of a minimal generator of ω (1−q) R . Using this, we obtain:…”