Open subsets of projective spaces with a good quotient by an action of a reductive group

Abstract

The aim of the paper is to describe all open subsets of a projective space with an action of a reductive group which admits a good quotient. As in the case of Mumford’s geometric invariant theory (which concerns projective good quotients) the problem can be reduced to the case of an action of a torus. We also show how to distinguish examples of open subsets with a good quotient coming from Mumford’s theory and give examples of open subsets with non-quasi-projective quotients.