GIT-equivalence beyond the ample cone

Details GIT-equivalence beyond the ample cone GIT-equivalence beyond the ample cone

2005-03-05

arxiv.org/abs/math/0503107v2

Michigan. Math. J., Vol. 54, No. 3, 483-516 (2006)

Mathematics

Algebraic Geometry

30 pages, minor corrections, to appear in Michigan Math. J

Scientific paper

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.

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