In both commutative algebra and algebraic geometry of positive characteristic, the study of Frobenius maps have led to significant developments. The theory of Frobenius splitting, introduced by Mehta-Ramanathan, is one of such developments. It is known that Frobenius-splitting varieties have good properties, for example, the Kodaira vanishing holds on such varieties. Furthermore, Frobenius-splitting has the simple criterion proved by Fedder. Yobuko recently introduced the notion of quasi-Fsplitting and F-split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that F-split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. In this talk, I will introduce a formula for F-split height and a criterion for quasi-F-splitting, which are generalizations of the Fedder's criterion. Moreover, I will talk about a relationship between the quasi-F-splitting property and pathological phenomenon in positive characteristic.
This talk is based on joint work with Tatsuro Kawakami and Teppei Takamatsu.