Products of Brauer-Severi surfaces

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 if and only if $\prod P_i$ is birational to $\prod Q_i$. Moreover in this case $\prod P_i$ and $\prod Q_i$ represent the same class in $M(k)$, the Grothendieck ring of $k$-varieties. The converse holds if $\mathrm {char}(k)=0$. Some of the above implications also hold over a general noetherian base scheme.