A simple proof for the existence of Zariski decompositions on surfaces

TL;DR

This paper presents a straightforward proof of the existence and uniqueness of Zariski decompositions on surfaces, introducing a constructive approach that simplifies the process and offers a practical algorithm for computation.

Contribution

It provides a simple, constructive proof of Zariski decompositions on surfaces, replacing complex procedures with a maximality-based approach.

Findings

Proof of existence and uniqueness of Zariski decompositions

A practical algorithm for computing the positive part

Simplification over original proofs

Abstract

In this note we give a quick and simple proof of the existence (and uniqueness) of Zariski decompositions on surfaces. While Zariski's original proof employs a rather sophisticated procedure to construct the negative part of the decomposition, the present approach is based on the idea that the positive part can be constructed from a maximality condition. It may also be useful that this approach yields a practical algorithm for the computation of the positive part.