Let k be a field and let G be an affine k-algebraic group. Call a Gtorsor weakly versal for a class of k-schemes C if it specializes to every G-torsor over a scheme in C . A recent result of the first author, Reichstein and Williams says that for any d ≥ 0, there exists a G-torsor over a finite type k-scheme that is weakly versal for finite type affine k-schemes of dimension at most d. The first author also observed that if G is unipotent, then G admits a torsor over a finite type k-scheme that is weakly versal for all affine k-schemes, and that the converse holds if char k = 0. In this work, we extend this to all fields, showing that G is unipotent if and only if it admits a G-torsor over a quasi-compact base that is weakly versal for all finite type regular affine k-schemes. Our proof is characteristic-free and it also gives rise to a quantitative statement: If G is a non-unipotent subgroup of GLn, then a G-torsor over a quasi-projective k-scheme of dimension d is not weakly versal for finite type regular affine kschemes of dimension n(d + 1) + 2. This means in particular that every such G admits a nontrivial torsor over a regular affine (n + 2)-dimensional variety. When G contains a nontrivial torus, we show that nontrivial torsors already exist over 3-dimensional smooth affine varieties (even when G is special), and this is optimal in general.

In the course of the proof, we show that for every m, ℓ ∈ N ∪ {0} with ℓ ̸ = 1, there exists a smooth affine k-scheme X carrying an ℓ-torsion line bundle that cannot be generated by m global sections. We moreover study the minimal possible dimension of such an X and show that it is m, m + 1 or m + 2.