## On Mori chamber and stable base locus decompositions

Antonio Laface
Abstract:
The effective cone of a Mori dream space [2] admits two decompositions into chambers called Mori chambers and stable base locus chambers (these are not necessarily convex cones). The former decomposition is a refinement of the latter, because Mori equivalent divisors have the same stable base locus. In this talk I will discuss both decompositions from a combinatorical viewpoint using the language of Cox rings [1,
$\S 3.3$
] and provide examples when the two are different. At the end I will state a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two.
This is joint work [3] with A. Massarenti and R. Rischter.
References
[1] Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015.
[2] Yi Hu and Sean Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
[3] Antonio Laface, Alex Massarenti, and Rick Rischter, On Mori chamber and stable base locus decompositionsarXiv:1805.10925.