On Mori chamber and stable base locus decompositions

TL;DR

This paper compares Mori chamber and stable base locus decompositions of the effective cone in Mori dream spaces, providing criteria for their coincidence and examples where they differ, especially in low Picard rank cases.

Contribution

It offers a criterion to determine when the two decompositions coincide in Picard rank two Mori dream spaces and constructs explicit examples where they differ.

Findings

The two decompositions coincide for certain Picard rank two Mori dream spaces.

Explicit examples show the decompositions can differ in Picard rank two.

Classification of smooth toric 3-folds with differing decompositions at Picard rank three.

Abstract

The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our criterion is sharp. Finally, we classify the smooth toric 3-folds of Picard rank three for which the two decompositions are different.