GIT-equivalence beyond the ample cone

TL;DR

This paper extends the understanding of GIT-equivalence beyond ample divisors for algebraic torus actions, providing a combinatorial framework and applications to exotic projective orbit spaces.

Contribution

It offers a combinatorial description of GIT-equivalence classes for non-ample divisors and extends properties of ample GIT-classes to a broader setting.

Findings

Develops a combinatorial description of GIT-equivalence classes.

Extends properties of ample GIT-classes to non-ample cases.

Provides an example illustrating systematic treatment of exotic projective orbit spaces.

Abstract

Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the partially ordered set of GIT-equivalence classes. As an application, we extend in the $\QQ$-factorial case a basic feature of the collection of ample GIT-classes to the partially ordered collection of maximal subsets with a quasiprojective quotient: for any two members there is at most one minimal member comprising both of them. Moreover, we demonstrate in an example, how our theory can be applied for a systematic treatment of ``exotic projective orbit spaces'', i.e., projective geometric quotients that do not arise from any linearized ample divisor.