Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Abstract

Let k be an algebraically closed field. Let P(X₁₁,..., X_nn, T) be the characteristic polynomial of the generic matrix (X_ij) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.