A birational interpretation of Severi-Brauer varieties

Let F be a field, A an F-csa of degree n. Saltman proves that there is a Zariski dense subset of elements d ϵ A such that the Severi-Brauer variety of A is birationally equivalent to the affine variety defined by restricting the reduced norm to subsets of the form K + d where K is a commutative separable maximal F subalgebra of A. In this work, we show that Saltman’s result actually implies that there is a non-empty dense subset of subspaces V≤A of dimension n + 1 such that the Severi-Brauer variety of A is birationally equivalent to the projective variety defined by restricting the reduced norm of A to V. We then show that for symbol algebras, a standard n-Kummer subspace induces the birational equivalence as above and use it to reprove Amitsur’s conjecture for the case of symbol algebras.