On Mori chamber and stable base locus decompositions

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.

Finite Generation of Cox Rings

“…3.3.2.9] Mov(X) = There are 17 full-dimensional Mori chambers, five of which are inside the moving cone. The nef cone is the black chamber, while the two gray chambers have the same stable base locus [LMR20].…”

On the Weak Lefschetz Principle in Birational Geometry

“…The following definition focuses on two possible forms of the weak Lefschetz principle in this context. They were introduced and explored in [13], [10]. We say that and are birational twins if they are Lefschetz divisorially equivalent Mori dream spaces and in addition * MCD( ) = MCD( ).…”

“…The following definition focuses on two possible forms of the weak Lefschetz principle in this context. They were introduced and explored in [LHM20], [LMR20].…”

Small modifications of Mori dream spaces arising from ${\mathbb C}^*$-actions

“…The Mori chamber decomposition of X . We will now describe the Mori chamber decomposition of the movable cone of X , which in general is only known to be a refinement of the stable base locus decomposition of the movable cone (see for instance [17,Remark 2.5]). In our case we will show: Theorem 4.10.…”