Paper
Algebraic Groups with Torsors That Are Versal for All Affine Varieties
Authors
Uriya A. First, Mathieu Florence, Zev Rosengarten
Abstract
Let be a field and let be an affine algebraic group over . Call a -torsor weakly versal for a class of -schemes if it specializes to every -torsor over a scheme in . A recent result of the first author, Reichstein and Williams says that for any , there exists a -torsor over a finite type -scheme that is weakly versal for finite type affine -schemes of dimension at most . The first author also observed that if is unipotent, then admits a torsor over a finite type -scheme that is weakly versal for all affine -schemes, and that the converse holds if . In this work, we extend this to all fields, showing that is unipotent if and only if it admits a -torsor over a quasi-compact base that is weakly versal for all finite type regular affine -schemes. Our proof is characteristic-free and it also gives rise to a quantitative statement: If is a non-unipotent subgroup of , then a -torsor over a quasi-projective -scheme of dimension is not weakly versal for finite type regular affine -schemes of dimension . This means in particular that every such admits a nontrivial torsor over a regular affine -dimensional variety. When contains a nontrivial torus, we show that nontrivial torsors already exist over -dimensional smooth affine varieties (even when is special), and this is optimal in general.
In the course of the proof, we show that for every with , there exists a smooth affine -scheme carrying an -torsion line bundle that cannot be generated by global sections. We moreover study the minimal possible dimension of such an and show that it is , or .