The flat locus of Brauer–Severi fibrations of smooth orders

Given a Cayley–Hamilton smooth order A in a central simple algebra Σ, 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 toric quiver varieties and use this fact to compute their cohomology.