On integral Zariski decompositions of pseudoeffective divisors on algebraic surfaces

TL;DR

This paper investigates when Zariski decompositions of pseudoeffective integral divisors on algebraic surfaces are integral, revealing that integrality can imply all negative curves are (-1)-curves, but exceptions exist.

Contribution

It provides new insights into the conditions under which Zariski decompositions are integral and explores the relationship with negative curves on algebraic surfaces.

Findings

Integrality sometimes implies all negative curves are (-1)-curves.

Examples show that integrality does not always force negative curves to be (-1)-curves.

The paper highlights the nuanced relationship between integrality and the negativity of curves.

Abstract

In this note we consider the problem of integrality of Zariski decompositions for pseudoeffective integral divisors on algebraic surfaces. We show that while sometimes integrality of Zariski decompositions forces all negative curves to be $(-1)$-curves, there are examples where this is not true.