Abstract
Given 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 Bia lynicki-Birula and Świȩcicka for varieties with semisimple group action from the smooth to the singular case.
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