Effective Matsusaka's Theorem for surfaces in characteristic p

TL;DR

This paper provides an effective bound for when an ample line bundle on smooth algebraic surfaces in positive characteristic becomes very ample, extending classical theorems with new bounds and vanishing results.

Contribution

It introduces an effective version of Matsusaka's theorem for surfaces in positive characteristic, including a Reider-type theorem for pathological cases.

Findings

Established an effective bound for very ampleness of line bundles

Proved a Kawamata-Viehweg-type vanishing theorem in positive characteristic

Extended classical theorems to include pathological surfaces in characteristic p

Abstract

We obtain an effective version of Matsusaka's theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple which makes an ample line bundle D very ample. The proof for pathological surfaces is based on a Reider-type theorem. As a consequence, a Kawamata-Viehweg-type vanishing theorem is proved for arbitrary smooth algebraic surfaces in positive characteristic.