First Steps towards Hyper-desingularization through Brauer-Severi Varieties

Details First Steps towards Hyper-desingularization through Brauer-Severi Varieties First Steps towards Hyper-desingularization through Brauer-Severi Varieties

2005-01-18

arxiv.org/abs/math/0501270v2

Mathematics

Algebraic Geometry

24 pages. Added missing reference. Made clear that throughout the paper we only work with smooth orders

Scientific paper

Given a Cayley-Hamilton smooth order A in a central simple algebra $\Sigma$, we determine the flat locus of the Brauer-Severi fibration of the smooth order. Moreover, we give a classification of all (reduced) central singularities where the flat locus differs from the Azumaya locus and show that the fibers over the flat, non-Azumaya points near these central singularities can be described as fibered products of graphs of projection maps, thus generalizing an old result of Artin on the fibers of the Brauer-Severi fibration over a ramified point. Finally, we show these fibers are also toric quiver varieties and use this fact to compute their cohomology.

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