The flat locus of Brauer-Severi fibrations of smooth orders.
For a Cayley-Hamilton algebra \(A\), let \(\text{trep}_n(A)\) denote the variety of trace preserving \(n\)-dimensional representations of \(A\), and let \(\text{triss}_n(A)\) be its quotient modulo the \(\text{GL}_n\)-action. If \(\text{BS}_n(A)\) denotes the trace preserving Brauer-Severi variety, there is a fibration \(\pi\colon\text{BS}_n(A)\to\text{triss}_n(A)\). The authors consider smooth orders \(A\), i.e.~orders in a central simple algebra for which \(\text{trep}_n(A)\) is a smooth variety. For such Cayley-Hamilton algebras \(A\), the étale local structure of the maps \(\text{BS}_n(A)\to\text{trep}_n(A)\to\text{triss}_n(A)\) can be described in terms of marked quivers. Using this method, the authors describe the flat locus of Brauer-Severi schemes of smooth orders. Furthermore, they analyse the singularities of \(\text{triss}_n(A)\) where the flat locus does not coincide with the Azumaya locus, and describe the fibers near such singularities. This generalizes classical results of M.~Artin.
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