Orbits of actions of group superschemes

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 $x$.

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