An integral version of Zariski decompositions on normal surfaces

TL;DR

This paper introduces an integral version of Zariski decompositions for normal surfaces, enabling new vanishing theorems, Reider-type results, and extension theorems for divisors and morphisms.

Contribution

It develops a novel integral decomposition of divisors on normal surfaces, extending classical Zariski decompositions and applying them to various geometric theorems.

Findings

Established a unique integral positive-negative divisor decomposition.

Derived a generalized vanishing theorem for divisors on surfaces.

Proved Reider-type theorems and extension results for morphisms.

Abstract

We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and Paoletti's results).