  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 the semidirect product
$P_{n,\tau}:=PGL_n\rtimes<\tau>$, where $\tau$ is the automorphism of $\PGLn$
given by $\tau(h)=(h^{-1})^transpose$. Basic properties of an involution are
described in terms of the action of $P_{n,\tau}$ on the associated variety, and
in particular in terms of the stabilizer in general position for this action.
