Geodesic
Severi-Brauer varieties of semidirect product algebras.
Documenta mathematica, Tome 8 (2003), pp. 527-546
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A conjecture of Amitsur states that two Severi-Brauer varieties are birationally isomorphic if and only if the underlying algebras are the same degree and generate the same cyclic subgroup of the Brauer group. It is known that generating the same cyclic subgroup is a necessary condition, however it has not yet been shown to be sufficient. In this paper we examine the case where the algebras have a maximal subfield K/F of degree n with Galois closure E/F whose Galois group is of the form Cn​⋊H, where EH=K and ∣H∣ is prime to n. For such algebras we show that the conjecture is true for certain cases of n and H. In particular we prove the conjecture in the case that G is a dihedral group of order 2p, where p is prime.
DOI : 10.4171/dm/149
Classification : 16K50, 16S35
Mots-clés : Brauer groups, Severi-Brauer varieties 
@article{10_4171_dm_149,
     author = {Daniel Krashen},
     title = {Severi-Brauer varieties of semidirect product algebras.},
     journal = {Documenta mathematica},
     pages = {527--546},
     year = {2003},
     volume = {8},
     doi = {10.4171/dm/149},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/149/}
}
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Daniel Krashen. Severi-Brauer varieties of semidirect product algebras.. Documenta mathematica, Tome 8 (2003), pp. 527-546. doi: 10.4171/dm/149

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