Smooth rational surfaces violating Kawamata-Viehweg vanishing

TL;DR

This paper demonstrates that in positive characteristic, there are smooth rational surfaces where Kawamata-Viehweg vanishing fails, challenging the expected vanishing behavior in algebraic geometry.

Contribution

It constructs explicit examples of smooth rational surfaces in positive characteristic that violate Kawamata-Viehweg vanishing, providing new insights into the limitations of vanishing theorems.

Findings

Existence of such surfaces over any algebraically closed field of positive characteristic

Counterexamples to Kawamata-Viehweg vanishing in dimension two

Implications for the behavior of vanishing theorems in positive characteristic

Abstract

We show that over any algebraically closed field of positive characteristic, there exists a smooth rational surface which violates Kawamata-Viehweg vanishing.