2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/212904We consider actions of reductive groups on a varieties 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 of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces.27 pages, minor changes, Example 8.8 replaced, to appear in Journal of Pure and Applied AlgebraAlgebraic Geometry14L24, 14L30, 14C20text