Rigidity And Unirational Groups

TL;DR

This paper proves a rigidity theorem for certain morphisms into solvable groups, leading to new structural insights about unirational group schemes and establishing properties of permawound unipotent groups.

Contribution

It introduces a rigidity theorem for morphisms into solvable groups and derives new structural results about unirational group schemes, including their behavior under separable extensions.

Findings

Unirationality descends through separable extensions.

Permawound unipotent groups are unirational.

Wound unipotent groups are commutative when unirational.

Abstract

We prove a rigidity theorem for morphisms from products of open subschemes of the projective line into solvable groups not containing a copy of $\Ga$ (for example, wound unipotent groups). As a consequence, we deduce several structural results about unirational group schemes, including that unirationality for group schemes descends through separable extensions. We also apply the main result to prove that permawound unipotent groups are unirational and -- when wound -- commutative.