Central simple algebras with involution: a geometric approach

Let k be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free PGLn-varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm PGL}_n\rtimes\langle\tau\rangle}$$\end{document}, where τ is the automorphism of PGLn given by τ (h) = (h−1)transpose. Basic properties of an involution are described in terms of the action of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm PGL}_n\rtimes\langle\tau\rangle}$$\end{document} on the associated variety, and in particular in terms of the stabilizer in general position for this action.