Highly versal torsors

Let G be a linear algebraic group over an infinite field k. Loosely speaking, a G-torsor over a k-variety is said to be versal if it specializes to every G-torsor over any k-field. The existence of versal torsors is well-known. We show that there exist G-torsors that admit even stronger versality properties. For example, for every d ∈ N, there exists a G-torsor over a smooth quasiprojective k-scheme that specializes to every torsor over a quasi-projective k-scheme after removing some codimension-d closed subset from the latter. Moreover, such specializations are abundant in a well-defined sense. Similar results hold if we replace k with an arbitrary base-scheme. In the course of the proof we show that every globally generated rank-n vector bundle over a d-dimensional k-scheme of finite type can be generated by n+d global sections. When G can be embedded in a group scheme of unipotent upper-triangular matrices, we further show that there exist G-torsors specializing to every G-torsor over any affine k-scheme. We show that the converse holds when char k=0. We apply our highly versal torsors to show that, for fixed m, n ∈ N, the symbol length of any degree-m period-n Azumaya algebra over any local Z[1 n, e2πi/n]-ring is uniformly bounded. A similar statement holds in the semilocal case, but under mild restrictions on the base ring.