A description of the Brauer-Severi scheme of trace rings of generic matrices

Here we use invariant theory to describe the Brauer-Severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants. Using this approach, we first prove that the Brauer-Severi scheme of a trace ring is isomorphic to Proj Q, for a graded ring Q whose generators we describe in the first section. This description also has a relevant interpretation over base fields of arbitrary characteristic. In the second section of this paper we show that the Brauer-Severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as Brauer-Severi schemes of certain algebras. (C) 1996 Academic Press, Inc.