Extensions Of Unirational Groups

TL;DR

The paper analyzes extensions of unirational algebraic groups over fields of positive characteristic. It proves that extensions preserve unirationality when the field has imperfection degree $1$, but provides explicit counterexamples for higher imperfection degrees, answering Achet’s question in the negative in those cases. It then introduces ext-unirational groups, establishing equivalences with centralizers of tori and with the maximal commutative $p$-torsion quotient, and shows that ext-unirationality passes through quotients and is birationally invariant. Finally, it relates unirationality of unipotent groups to quotients by their lower central series, showing that for imperfection degree up to $2$, unirationality is determined by $U^{ m ab}$, with sharp counterexamples for larger imperfection degrees. The results collectively clarify when unirationality is preserved by extensions and provide a framework based on filtrations and quotients for analyzing unirationality in imperfect fields.

Abstract

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.

Theorems & Definitions (43)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Lemma 3.1
• proof