群超方案的作用轨道
Orbits of actions of group superschemes
摘要：
在代数闭域$\Bbbk$上，证明了代数群超方案$G$在有限型超方案$X$上的左作用的所有轨道都是局部闭的.此外，当且仅当$G_{ev}x$在$X_{ev}$中闭合时，或者等效地，当且仅当$G_{res}x$在$X_{res}$中闭合时，这样的轨道$Gx$（其中$x$是$X$的$\Bbbk$ -点）是闭合的。这里，$G_{ev}$是$G$的最大纯偶群超子方案，而$G_{res}$是被视为群方案的$G_{ev}$。类似地，$X_{ev}$是$X$的最大纯偶超子方案，而$X_{res}$是被视为方案的$X_{ev}$。我们还证明了$\mathrm{sdim}(Gx)=\mathrm{sdim}(G)-\mathrm{sdim}(G_x)$，其中$G_x$是$x$的稳定剂。
英文摘要：
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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