Orbits of actions of group superschemes

Abstract

Working over an algebraically closed field k, we prove that all orbits of a left action of an algebraic group superscheme G on a superscheme X of finite type are locally closed. Moreover, such an orbit Gx, where x is a k-point of X, is closed if and only if Gevx is closed in Xev, or equivalently, if and only if Gresx is closed in Xres. Here Gev is the largest purely even group super-subscheme of G and Gres is Gev regarded as a group scheme. Similarly, Xev is the largest purely even super-subscheme of X and Xres is Xev regarded as a scheme. We also prove that sdim(Gx)=sdim(G)−sdim(Gx), where Gx is the stabilizer of x.