Orbits of actions of group superschemes

TL;DR

This paper proves that all orbits of algebraic group superschemes acting on superschemes are locally closed, characterizes when such orbits are closed, and establishes a dimension formula relating orbits and stabilizers.

Contribution

It extends classical orbit closure and dimension results to the setting of algebraic group superschemes, providing new structural insights.

Findings

All orbits are locally closed.

Orbit closure characterized by even parts.

Dimension formula for orbits and stabilizers.

Abstract

Working over an algebraically closed field $\Bbbk$, 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 $\Bbbk$-point of $X$, is closed if and only if $G_{ev}x$ is closed in $X_{ev}$, or equivalently, if and only if $G_{res}x$ is closed in $X_{res}$. Here $G_{ev}$ is the largest purely even group super-subscheme of $G$ and $G_{res}$ is $G_{ev}$ regarded as a group scheme. Similarly, $X_{ev}$ is the largest purely even super-subscheme of $X$ and $X_{res}$ is $X_{ev}$ regarded as a scheme. We also prove that $\mathrm{sdim}(Gx)=\mathrm{sdim}(G)-\mathrm{sdim}(G_x)$, where $G_x$ is the stabilizer of $x$.