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_{e...

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Bibliographic Details
Published in:arXiv.org 2022-02
Main Authors: Bovdi, V A, Zubkov, A N
Format: Article
Language:English
Subjects:
Orbits
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Summary: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 \(x\).
ISSN:2331-8422