Highly Versal Torsors

Let $G$ be a linear algebraic group over an infinite field $k$. Loosely speaking, a $G$-torsor over $k$-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 $d\in\mathbb{N}$, there exists a $G$-torsor over a smooth quasi-projective $k$-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 $\operatorname{char} k=0$. We apply our highly versal torsors to show that, for fixed $m,n\in\mathbb{N}$, the symbol length of any degree-$m$ period-$n$ Azumaya algebra over any local $\mathbb{Z}[\frac{1}{n},e^{2\pi i/n}]$-ring is uniformly bounded. A similar statement holds in the semilocal case, but under mild restrictions on the base ring.