dkrashen@math.ucla.edu

Submission: 2002, Mar 12

The aim of this paper is to investigate the birational geometry of Generalized Severi-Brauer varieties. A conjecture of Amitsur states that two Severi-Brauer varieties $V(A)$ and $V(B)$ are birational if the underlying algebras $A$ and $B$ are the same degree and generate the same cyclic subgroup of the Brauer group. We present a generalization of this conjecture to Generalized Severi-Brauer varieties, and show that in many cases we may reduce the new conjecture to the case where every subfield of the algebras is maximal, and in particular to the case where the algebras have prime power degree. This allows us to prove infinitely many new cases of Amitsur's original conjecture. We also give a proof of the generalized conjecture for the case $B \cong A^{op}$.

2000 Mathematics Subject Classification: 16K20, 16K50, 14E05

Keywords and Phrases: central simple algebra, Severi-Brauer variety, generalized Severi-Brauer variety, Brauer group, division algebra

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