Fibers of generic Brauer-Severi schemes
Brauer-Severi varieties were introduced in 1944 by F. Châtelet to obtain a deeper understanding of the relation between a central simple algebra and the splitting fields of the algebra. Brauer-Severi schemes were introduced by M. Van den Bergh to study a wider class of algebras. In particular, he defined the Brauer-Severi schemes for the free algebra with \(m\) generators over a commutative ring \(R\) which the authors of the paper under review call generic Brauer-Severi schemes and show that they can be realized as moduli spaces of representations of a specific quiver. For an order in a central simple algebra, the Brauer-Severi scheme is a projective fiber bundle over the variety of the centre. As the generic Brauer-Severi scheme is in a sense the most general one, it is important to study the geometry of its fibers and to compute their dimensions in order to determine the flat locus. In the paper under review the authors continue the study of the local structure of generic Brauer-Severi schemes started in a recent work by one of the authors. The main result is the following. For the Brauer-Severi scheme \(\eta\colon BS_{m,n}\to V_{m,n}\) of the trace ring of \(m\) generic \(n\times n\) matrices over the variety of matrix invariants, for a point \(\xi\in V_{m,n}\) of representation type \((m_1,n_1;\ldots;m_r,n_r)\), the reduced variety of \(\eta^{-1}(\xi)\) has \((\sum m_i)!/\prod m_i!\) irreducible components, each of the same explicitly given dimension.
moduli spaces of representations of quivers
scientific article; zbMATH DE number 1275262
scientific article