Categorical quotients of certain algebraic group actions
Let k be an algebraically closed field, H a connected unipotent algebraic group and X a normal quasi-affine k-variety on which H acts with finite stabilizers. A theorem of \textit{C. S. Seshadri} [Ann. Math., II. Ser. 95, 511-556 (1972; Zbl 0241.14024) and 96, 599 (1972)] gives the existence of a finite H-morphism \(Z\to X\) where H acts locally trivially on Z, and hence the geometric quotient W of Z by H exists. W is an algebraic pre- scheme, and in general not separated. The author terms W ''almost quasi- affine'' if there is a quasi-finite surjective morphism \(f:W\to Y,\) where Y is a quasi-affine variety. Since X is normal, Z and W can be taken normal and in this case f can be chosen birational and surjective and Y is essentially unique. Seshadri's construction actually exhibits X as a quotient of Z by a finite group G of H-equivariant automorphisms. G acts on W and hence also on Y, and the quotient of Y by G is, in a sense, a quotient of X by H: the sense is in the category of separated algebraic k-schemes. - The author also obtains a generalization of this result to non-unipotent groups H, under the assumption that X has a factorial finitely generated ring of global sections.
scientific article; zbMATH DE number 3851277