Weyl and Zariski chambers on K3 surfaces

TL;DR

This paper compares two natural chamber decompositions of the big cone on K3 surfaces, analyzing their relationship, conditions for coincidence, and showing they always have the same number of chambers despite possible differences.

Contribution

It provides a numerical criterion for the coincidence of Zariski and Weyl chambers and proves they always have equal chamber counts on K3 surfaces.

Findings

Numerical criterion for chamber coincidence

Mutual inclusion relations between chambers

Equal number of Zariski and Weyl chambers

Abstract

The big cone of every K3 surface admits two natural chamber decompositions: the decomposition into Zariski chambers, and the decomposition into simple Weyl chambers. In the present paper we compare these two decompositions and we study their mutual relationship: First, we give a numerical criterion for the two decompositions to coincide. Secondly, we study the mutual inclusions of Zariski and simple Weyl chambers. Finally, we establish the fact that -- even though the decompositions themselves may differ -- the number of Zariski chambers always equals the number of simple Weyl chambers.