Rational maps between varieties associated to central simple algebras

TL;DR

This paper investigates rational maps between varieties linked to central simple algebras, revealing new connections when these algebras generate the same subgroup in the Brauer group, and relates findings to the Amitsur conjecture.

Contribution

It establishes conditions for the existence of rational maps and embeddings between varieties associated to central simple algebras based on their Brauer group relations.

Findings

Rational maps exist between associated varieties when algebras generate the same subgroup in Br(k)

In some cases, rational embeddings are possible between these varieties

Results provide insights related to the Amitsur conjecture

Abstract

In this paper we show that if two central simple $k$-algebras generate the same cyclic subgroup in $\mathrm{Br}(k)$, then there are rational maps between varieties associated to these algebras, such as Brauer--Severi varieties, norm hypersurfaces and symmetric powers. In some cases we even have rational embeddings. We also relate the obtained results to the Amitsur conjecture.