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 PGL _n -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 , where τ is the automorphism of PGL _n given by τ (h) = (h ⁻¹)^transpose. Basic properties of an involution are described in terms of the action of on the associated variety, and in particular in terms of the stabilizer in general position for this action.