Limits of rank 4 Azumaya algebras and applications to desingularization

It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes Seshadri's theorem in [16] that the variety of specializations of (2 x 2)-matrix algebras is smooth in characteristic not equal 2. As an application, a construction of Seshadri in [16] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of Nori over Z (Appendix, [16]) is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [1] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property if the Arlin moduli space has geometrically reduced fibers-for example this happens over Z. Essential use is made of Kneser's concept [8] of 'semi-regular quadratic module'. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived.