Zariski decomposition: a new (old) chapter of linear algebra

TL;DR

This paper reinterprets Zariski decomposition within linear algebra, providing a new theorem and proof that connect algebraic geometry concepts to linear algebraic frameworks.

Contribution

It formulates Zariski decomposition as a linear algebra theorem and offers a linear algebraic proof, bridging geometry and algebra.

Findings

Zariski decomposition can be expressed purely in linear algebra terms

A new linear algebraic proof of Zariski decomposition is provided

The geometric origins of Zariski decomposition are briefly discussed

Abstract

In a 1962 paper, Zariski introduced the decomposition theory that now bears his name. Although it arose in the context of algebraic geometry and deals with the configuration of curves on an algebraic surface, we have recently observed that the essential concept is purely within the realm of linear algebra. In this paper, we formulate Zariski decomposition as a theorem in linear algebra and present a linear algebraic proof. We also sketch the geometric context in which Zariski first introduced his decomposition.