Invariants of finite group schemes

Let k be an algebraically closed field, G a finite group scheme over k operating on a scheme X over k. Under assumption that X can be covered by G-invariant affine open subsets the classical results in [3] and [14] describe the quotient X/G. In case of a free action X is known to be a principal homogeneous G-space over X/G. Furthermore, the category of G-linearized quasi-coherent sheaves of OX -modules is equivalent then to the category of quasi-coherent sheaves of OX/G-modules. In this paper we attempt to describe the situation when generic stabilizers of points on X are nontrivial. To avoid technical complications we assume that X is an algebraic variety, although the results can be extended to reduced schemes. The stabilizer Gx of a rational point x ∈ X is a subgroup scheme of G, and we define its index (G : Gx) by analogy with the ordinary finite groups. A point x is regular with respect to the action of G if the index (G : Gx) attains the maximal possible value q(X). Theorem 2.1 shows that the set XG-reg of all regular points is an open G-invariant subset of X , the restriction to which of the canonical morphism π : X → X/G is finite flat of degree q(X). For every x ∈ XG-reg the fibre π (