Let
G
G
be a linear algebraic group over an infinite field
k
k
. Loosely speaking, a
G
G
-torsor over a
k
k
-variety is said to be versal if it specializes to every
G
G
-torsor over any
k
k
-field. The existence of versal torsors is well-known. We show that there exist
G
G
-torsors that admit even stronger versality properties. For example, for every
d
∈
N
d\in \mathbb {N}
, there exists a
G
G
-torsor over a smooth quasi-projective
k
k
-scheme that specializes to every torsor over a quasi-projective
k
k
-scheme after removing some codimension-
d
d
closed subset from the latter. Moreover, such specializations are abundant in a well-defined sense. Similar results hold if we replace
k
k
with an arbitrary base-scheme. In the course of the proof we show that every globally generated rank-
n
n
vector bundle over a
d
d
-dimensional
k
k
-scheme of finite type can be generated by
n
+
d
n+d
global sections.
When
G
G
can be embedded in a group scheme of unipotent upper-triangular matrices, we further show that there exist
G
G
-torsors specializing to every
G
G
-torsor over any affine
k
k
-scheme. We show that the converse holds when
c
h
a
r
k
=
0
chark=0
.
We apply our highly versal torsors to show that, for fixed
m
,
n
∈
N
m,n\in \mathbb {N}
, the symbol length of any degree-
m
m
period-
n
n
Azumaya algebra over any local
Z
[
1
n
,
e
2
π
i
/
n
]
\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.