Extensions Of Unirational Groups

We undertake a study of extensions of unirational algebraic groups. We prove that extensions of unirational groups are also unirational over fields of degree of imperfection $1$, but that this fails over every field of higher degree of imperfection, answering a question of Achet. We also initiate a study of those groups which admit filtrations with unirational graded pieces, and show that one may deduce unirationality of unipotent groups from unirationality of certain quotients.

💡 Research Summary

The paper investigates the behavior of unirational algebraic groups under extensions, focusing on the role of the field’s degree of imperfection. The main results can be summarized as follows.

1. Background and Motivation. Over perfect fields every connected linear algebraic group is unirational, but this fails in the imperfect setting. A question posed by Acet asks whether a commutative extension of two unirational groups must be unirational. The answer is known to be “yes’’ over perfect fields and “trivially’’ when one of the groups is split, but remained open for general imperfect fields.

2. Perma‑wound groups and degree‑1 fields. The authors recall the notion of a perma‑wound unipotent group (introduced in earlier work). A smooth unipotent group (U) over a field (K) is perma‑wound if for every exact sequence (1\to U\to E\to G_a\to1) with (G_a) a finite‑type (p)-torsion group, the map (G_a\to E) admits a section. Using results from