Birationality for faithful twisted parabolic varieties over imperfect fields

Determine whether two twisted parabolic varieties over an imperfect perfect? field—specifically, twisted parabolic varieties X_φ and X_ψ with the same underlying reduced parabolic subgroups over an algebraic closure and with faithful G-actions—are birational as k-varieties.

Background

Question \ref{q:birational} asks whether the twisting data encoded by Wenzel functions affect birationality when the corresponding reduced parabolic subgroups agree. The paper gives a negative answer over imperfect fields when faithfulness of the G-action is not imposed, using a Severi–Brauer variety and a projective space obtained through a Frobenius quotient.

The authors leave unresolved whether the negative phenomenon persists under the additional requirement that the G-action be faithful. This problem concerns the birational classification of twisted parabolic varieties beyond their reduced geometric stabilizers.

References

Still, it is not known whether the answer to the above question would be affirmative if we restrict to twisted parabolic varieties with faithful $G$-action when $k$ is imperfect.

— Complete homogeneous varieties over a field  (2610.21497 - Brion et al., 15 Oct 2026) in Remark 6.?, immediately following Question 6.?, Section 6, “Twisted parabolic varieties over an arbitrary field”

In fact, it is not clear whether the answer to the above question is affirmative over perfect fields even if $X_{\varphi}$ and $X_{\psi}$ are isomorphic (as varieties forgetting the $G$-action) to classical twisted flag varieties for some (not necessarily the same) groups.

— Complete homogeneous varieties over a field  (2610.21497 - Brion et al., 15 Oct 2026) in Remark 6.?, following Question 6.?, Section 6, “Twisted parabolic varieties over an arbitrary field”