F-THRESHOLDS OF GRADED RINGS

The a-invariant, the F-pure threshold, and the diagonal Fthreshold are three important invariants of a graded K-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly F-regular rings. In this article, we prove that these relations hold only assuming that the algebra is F-pure. In addition, we present an interpretation of the a-invariant for F-pure Gorenstein graded K-algebras in terms of regular sequences that preserve F-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition Sk. We also present analogous results and questions in characteristic zero.