Products of Brauer Severi surfaces

Details Products of Brauer Severi surfaces Products of Brauer Severi surfaces

2007-06-23

arxiv.org/abs/0706.3447v1

Mathematics

Algebraic Geometry

7 pages

Scientific paper

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.

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