First Steps towards Hyper-desingularization through Brauer-Severi Varieties

Bocklandt, Raf; Symens, Stijn; Van de Weyer, Geert

First Steps towards Hyper-desingularization through Brauer-Severi Varieties

Authors

Bocklandt, Raf

Symens, Stijn

Van de Weyer, Geert

Description

Given a Cayley-Hamilton smooth order A in a central simple algebra $Σ$, 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.
24 pages. Added missing reference. Made clear that throughout the paper we only work with smooth orders