The flat locus of Brauer-Severifibrations of smooth orders

Given a Cayley-Hamilton smooth order A in a central simple algebra Sigma, we determine the flat locus of the Brauer-Severi fibration of A. 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. This generalizes an old result of Artin on the fibers of the Brauer-Severi fibration of a maximal order over a ramified point. Finally, we show these fibers are also ionic quiver varieties and use this fact to compute their cohomology. (c) 2005 Elsevier Inc. All rights reserved.