When are Zariski chambers numerically determined?

TL;DR

This paper provides a criterion to determine when Zariski chambers on smooth projective surfaces are numerically characterized by Weyl chambers, extending known results from K3 surfaces to a broader class.

Contribution

It introduces a simple criterion for the interiors of Zariski chambers to be numerically determined Weyl chambers, generalizing Bauer-Funke's results from K3 surfaces to all smooth projective surfaces.

Findings

Criterion for Zariski chambers to be Weyl chambers

Extension of Bauer-Funke's results to arbitrary surfaces

Analysis of decompositions related to elliptic fibrations on Enriques surfaces

Abstract

The big cone of every smooth projective surface $X$ admits the natural decomposition into Zariski chambers. The purpose of this note is to give a simple criterion for the interiors of all Zariski chambers on $X$ to be numerically determined Weyl chambers. Such a criterion generalizes the results of Bauer-Funke on K3 surfaces to arbitrary smooth projective surfaces. In the last section, we study the relation between decompositions of the big cone and elliptic fibrations on Enriques surfaces.