Products of Brauer Severi surfaces

TL;DR

The paper investigates the relationship between products of Brauer Severi surfaces and conics over a field, establishing conditions under which their generated subgroups in the Brauer group coincide and their birational equivalence.

Contribution

It proves that the subgroup generated by collections of Brauer Severi surfaces or conics is the same if and only if their products are birational, linking algebraic and birational properties.

Findings

Generated subgroups in Br(k) coincide iff products are birational.

Birational equivalence implies same class in Grothendieck ring.

Results extend to schemes under certain conditions.

Abstract

Let $\{P_i\}_{1 \leq i \leq r}$ and $\{Q_i\}_{1 \leq i \leq r}$ be two collections of Brauer Severi surfaces (resp. conics) over a field $k$. We show that the subgroup generated by the $P_i's$ in $Br(k)$ is the same as the subgroup generated by the $Q_i's$ \iff $\Pi P_i $ is birational to $\Pi Q_i$. Moreover in this case $\Pi P_i$ and $\Pi Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $char(k)=0$. Some of the above implications also hold over a general noetherian base scheme.