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 PGL_n× 〈τ〉, 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 PGL_n× 〈τ〉 on the associated variety, and in particular in terms of the stabilizer in general position for this action.