Algebraic Groups with Torsors That Are Versal for All Affine Varieties

Let $k$ be a field and let $G$ be an affine algebraic group over $k$. Call a $G$-torsor weakly versal for a class of $k$-schemes $\cal C$ if it specializes to every $G$-torsor over a scheme in $\cal C$. A recent result of the first author, Reichstein and Williams says that for any $d\geq 0$, 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 $\operatorname{char} k=0$. 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 $\mathbf{GL}_n$, then a $G$-torsor over a quasi-projective $k$-scheme of dimension $d$ is not weakly versal for finite type regular affine $k$-schemes 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 $m,\ell\in\mathbb{N}\cup\{0\}$ with $\ell\neq 1$, there exists a smooth affine $k$-scheme $X$ carrying an $\ell$-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 $m+2$.