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

Anurag K. Singh, Shunsuke Takagi, Matteo Varbaro

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.

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

Paper Structure

This paper contains 5 sections, 32 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $F$-singularities
  3. Positive characteristic case I
  4. Positive characteristic case II
  5. Characteristic zero case

Key Result

Lemma 2.4

Let $(R, \mathfrak{m})$ be a $d$-dimensional $F$-finite normal local ring of characteristic $p>0$, $\Delta$ be an effective $\mathbb{Q}$-divisor on $X=\operatorname{Spec} R$ and $\mathfrak{a}$ be a nonzero ideal of $R$. For any real number $t \geqslant 0$, the pair $((R, \Delta); \mathfrak{a}^t)$ is is injective, where $F^e_{X, \Delta}$ is the map induced by the $R$-linear map $R \longrightarrow F

Theorems & Definitions (84)

  • Conjecture 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: cf. HW
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 74 more