Counting Zariski chambers on Del Pezzo surfaces

TL;DR

This paper introduces an algorithm to compute Zariski chambers on algebraic surfaces, specifically applied to Del Pezzo surfaces, enhancing understanding of their linear series structure.

Contribution

The paper presents a novel algorithm for determining Zariski chambers based on negative curves, with applications to Del Pezzo surfaces.

Findings

Algorithm effectively computes Zariski chambers.

Number of chambers on Del Pezzo surfaces determined.

Provides insights into the structure of linear series on surfaces.

Abstract

Zariski chambers provide a natural decomposition of the big cone of an algebraic surface into rational locally polyhedral subcones that are interesting from the point of view of linear series. In the present paper we present an algorithm that allows to effectively determine Zariski chambers when the negative curves on the surface are known. We show how the algorithm can be used to compute the number of chambers on Del Pezzo surfaces.