The GIT-Equivalence for G-Line Bundles

Let X be a projective variety with an action of a reductive group G. Each ample G-line bundle L on X defines an open subset X^ss(L) of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of L's with fixed X^ssL. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the G-ample cone. We also study the corresponding variations of quotients X^ss(L)//G. This sharpens results of Thaddeus and Dolgachev-Hu.