Quasi-$F$-splitting and smooth weak del Pezzo surfaces in mixed characteristic

TL;DR

This paper introduces quasi-$F$-splitting in mixed characteristic and demonstrates its applications to Kodaira-type vanishing theorems on certain algebraic surfaces, advancing understanding in algebraic geometry.

Contribution

It defines quasi-$F$-splitting in mixed characteristic and applies it to prove vanishing theorems for lifts of RDP del Pezzo surfaces, a novel approach in the field.

Findings

Established Kodaira-type vanishing for lifts of RDP del Pezzo surfaces.

Introduced the concept of quasi-$F$-splitting in mixed characteristic.

Extended vanishing results to new classes of algebraic surfaces.

Abstract

We introduce the notion of quasi-$F$-splitting in mixed characteristic and study Kodaira-type vanishing on quasi-$F$-splitting varieties. As an application, we prove a Kodaira-type vanishing on lifts of rational double point (RDP) del Pezzo surfaces.