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Products of Brauer-Severi surfaces. (English) Zbl 1161.14008
It has been conjectured that two Brauer-Severi varieties $$P,Q$$, over a field $$k$$, are birational if and only if they generate the same subgroup of the Brauer group Br($$k$$). The conjecture is known to be true for surfaces, but yet unproved in general. The author studies the situation for products of Brauer-Severi surfaces. The main result shows that if $$\{P_i\}$$ and $$\{Q_i\}$$ are two finite collections of Brauer-Severi surfaces, then $$\prod P_i$$ is birational to $$\prod Q_i$$ if and only if the two collections generate the same subgroup of Br($$k$$).
The proof is obtained by induction on the number of elements in the collections. It needs, at some step, to consider the case of Brauer-Severi varieties defined over a general noetherian base scheme.

##### MSC:
 1.4e+06 Rational and birational maps
##### Keywords:
Brauer-Severi varieties
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##### References:
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