Adequate Moduli Spaces and Geometrically Reductive Group Schemes

We introduce the notion of an adequate moduli space. The theory of adequate moduli spaces provides a framework for studying algebraic spaces which geometrically approximate algebraic stacks with reductive stabilizers in characteristic p. The definition of an adequate moduli space generalizes the existing notion of a good moduli space to characteristic p (and mixed characteristic). The most important examples of an adequate moduli space are: (1) the morphism from the quotient stack [X/G] of the semistable locus to the GIT quotient X//G and (2) the morphism from an algebraic stack with finite inertia to the Keel−Mori coarse moduli space. It is shown that most of the fundamental properties of the GIT quotient X//G follow from only the defining properties of an adequate moduli space. We provide applications of adequate moduli spaces to the structure of geometrically reductive and reductive group schemes. In particular, results of Seshadri and Waterhouse are generalized. The theory of adequate moduli spaces provides the possibility for intrinsic constructions of projective moduli spaces in characteristic p.