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The $a$ -invariant, the $F$ -pure threshold, and the diagonal $F$ -threshold 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 $S_{k}$ . We also present analogous results and questions in characteristic zero.



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