The GIT fan for a Mori dream space and the μ-secondary polytope

Abstract

Geometric invariant theory (GIT) was developed by Mumford as a method for constructing quotients by group actions in the context of algebraic geometry. This construction is not canonical and depends on a choice. A way to organize the different choices and their resulting quotients we use a combinatorial object known as the GIT fan. Mori dream spaces were introduced by Hu and Keel as projective varieties whose cone of effective divisors can be decomposed into a finite number of convex polyhedral cones called Mori chambers. As the name suggests they are ideal spaces for Mori’s Minimal Model Program, which deals with the classification of varieties up to birational equivalence. It was shown by Hu and Keel that a Mori dream space is naturally a GIT quotient of an affine variety by an algebraic torus. As a consequence, the GIT fan coincides with the Mori chamber decomposition. Thus, the birational geometry of a Mori dream space is an instance of variation of geometric invariant theory quotients (VGIT). In this work we develop a criterion to describe the GIT fan associated to a Mori dream space. This is done by using tropical geometry, toric geometry, and triangulations of marked polytopes. As a result of this approach, we define a polytope called the µ-secondary polytope whose normal fan is the GIT fan, generalizing the result in toric geometry.