2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66292Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's Geometric Invariant Theory. We obtain several new Hilbert-Mumford type theorems, and we extend a projectivity criterion of Bialynicki-Birula and Swiecicka for varieties with semisimple group action from the smooth to the singular case.Final version, to appear in Compositio MathAlgebraic Geometry14L24, 14L30text