Involutions and Brauer-Severi varieties (Q1815082)

The authors explicitly construct for any finite dimensional central simple algebra \(A\) over a fixed field \(F\) new central simple algebras \(s^2A\) and \(\lambda^2A\), which are Brauer equivalent to the product \(A\otimes A\). With this construction, they then derive, in a characteristic free context, a correspondence between involutions (of the first kind) on \(A\) and rational points on the Brauer-Severi varieties of \(s^2A^0\) and \(\lambda^2A^0\). More precisely, if \(BS_1\) denotes the ``usual'' Brauer-Severi variety, then the main result in this paper states that there is a one-to-one correspondence between involutions of orthogonal (resp. symplectic) type on \(A\) and the rational points in an open subset \({\mathcal O}\subseteq BS_1(s^2A^0)\) (resp. \({\mathcal S}\subseteq BS_1(\lambda^2A^0)\)). The open set \(\mathcal O\) is always dense, whereas \(\mathcal S\) is dense if the degree of \(A\) is even, and empty, when it is odd.