Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic

TL;DR

This paper proves a vanishing theorem for divisors on normal surfaces in positive characteristic, enabling new results on morphisms and rational points over arbitrary fields.

Contribution

It establishes the Kawamata-Viehweg vanishing theorem in positive characteristic and applies it to derive Reider-type and extension theorems for normal surfaces.

Findings

Proved Kawamata-Viehweg vanishing theorem for certain divisors

Derived Reider-type theorems for normal surfaces

Characterized rational points on plane curves over arbitrary fields

Abstract

We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are established. As an application of the extension theorems, we characterize non-singular rational points on any plane curve over an arbitrary base field in terms of rational functions on the curve.