Rational maps between varieties associated to central simple algebras

In this paper we show that there exist rational maps between varieties associated to isomorphism classes of central simple algebras such as Severi-Brauer varieties, norm hypersurfaces and symmetric powers, provided the Brauer-equivalence classes of the algebras involved generate the same cyclic subgroup in Br(k). In some cases we even get rational embeddings. We relate the obtained results to the Amitsur conjecture.