The F-pure threshold versus the a-invariant for standard graded rings

TL;DR

This paper proves a conjecture relating the F-pure threshold to the Gorenstein property in standard graded rings and extends it to section rings of F-split projective varieties, using geometric methods.

Contribution

It completes the proof of a conjecture connecting F-pure thresholds and Gorenstein rings and extends the result to broader classes of algebraic varieties.

Findings

Confirmed the conjecture for standard graded strongly F-regular rings.

Extended the conjecture to section rings of F-split projective varieties.

Utilized geometric techniques involving Proj of graded rings.

Abstract

Hirose, Watanabe and Yoshida conjectured a criterion for a standard graded strongly $F$-regular ring to be Gorenstein in terms of the $F$-pure threshold. We complete the proof of this conjecture. We also prove natural extensions of the conjecture to section rings of normal, $F$-split projective varieties with respect to globally generated ample divisors. Our proof exploits the geometry of the Proj of the graded ring.