On the Kawamata-Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic

TL;DR

This paper proves the Kawamata-Viehweg vanishing theorem for del Pezzo surfaces in positive characteristic and explores its implications for threefold singularities, demonstrating the theorem's sharpness with counterexamples.

Contribution

It establishes the vanishing theorem for certain surfaces in positive characteristic and links it to rationality of singularities, extending known results.

Findings

Proves Kawamata-Viehweg vanishing for del Pezzo surfaces in characteristic p>5

Shows klt threefold singularities are rational in this setting

Provides counterexamples in characteristic 5 confirming sharpness

Abstract

We prove the Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic $p>5$. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic $p>5$ are rational. We show that these theorems are sharp by providing counterexamples in characteristic $5$.