Weyl and Zariski chambers on projective surfaces

TL;DR

This paper investigates the relationships between Weyl and Zariski chambers on complex projective surfaces, analyzing their decompositions of the big cone and conditions for their intersections and containments.

Contribution

It provides criteria for when Weyl chambers are contained in or intersect with Zariski chambers on nonsingular complex projective surfaces.

Findings

Weyl chambers can be contained in Zariski chambers under certain conditions.

Conditions are established for non-trivial intersections between Weyl and Zariski chambers.

The study clarifies the geometric structure of the big cone decompositions.

Abstract

Let $X$ be a nonsingular complex projective surface. The Weyl and Zariski chambers give two interesting decompositions of the big cone of $X$. We study these two decompositions and determine when a Weyl chamber is contained in the interior of a Zariski chamber and vice versa. We also determine when a Weyl chamber can intersect non-trivially with a Zariski chamber.