Geometric invariant theory via Cox rings

We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand–MacPherson type correspondences relating quotients by reductive groups to quotients by torus actions. Moreover, our approach provides a general access to the geometry of many of the resulting quotient spaces.