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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.

14E05 Rational and birational maps
Full Text: DOI arXiv
[1] S. A. Amitsur, Generic splitting fields of central simple algebras, Ann. of Math. (2) 62 (1955), 8 – 43. · Zbl 0066.28604
[2] János Kollár, Conics in the Grothendieck ring, Adv. Math. 198 (2005), no. 1, 27 – 35. · Zbl 1082.14022
[3] Michael Larsen and Valery A. Lunts, Motivic measures and stable birational geometry, Mosc. Math. J. 3 (2003), no. 1, 85 – 95, 259 (English, with English and Russian summaries). · Zbl 1056.14015
[4] Peter Roquette, On the Galois cohomology of the projective linear group and its applications to the construction of generic splitting fields of algebras, Math. Ann. 150 (1963), 411 – 439. · Zbl 0114.02206
[5] S. L. Tregub, Birational equivalence of Brauer-Severi manifolds, Uspekhi Mat. Nauk 46 (1991), no. 6(282), 217 – 218 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 6, 229. · Zbl 0785.14005
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