Birational isomorphisms between generalized Severi-Brauer varieties

Abstract

The aim of this paper is to investigate the birational geometry of Generalized Severi-Brauer varieties. A conjecture of Amitsur states that two Severi-Brauer varieties $V(A)$ and $V(B)$ are birational if the underlying central simple algebras $A$ and $B$ are the same degree and generate the same cyclic subgroup of the Brauer group. We present a generalization of this conjecture to Generalized Severi-Brauer varieties, and show that in most cases we may reduce the new conjecture to the case where every subfield of the algebras is maximal, and in particular to the case where the algebras have prime power degree. This allows us to prove infinitely many new cases for Amitsur's original conjecture. We also give a proof of the generalized conjecture for the case $B \cong A^{op}$.