Complete homogeneous varieties over a field

Abstract: The main objects of this paper are complete homogeneous varieties under the action of a semisimple group $G$ over field $k$ of characteristic $p>0$. When the isotropy groups of the geometric points are smooth parabolics, these are classically known as twisted flag varieties and are well studied in the literature. If the smoothness assumption is removed, we get a larger class of varieties which we call as \emph{twisted parabolic varieties}. The main goal of this paper is to study the structure, classification and properties of these varieties. We first understand their structure over a separable closure of $k$ by classifying non-smooth parabolics containing a Borel; this builds on the recently completed classification of such parabolics over an algebraically closed field. Then, via Galois descent we get a combinatorial criterion for such a variety to descend over $k$. We compute the Picard group of twisted parabolic varieties by embedding them into products of twisted flag varieties, study their local rigidity properties and show that they are rational if they have a $k$-rational point for $p \geq 3$. When $k$ is perfect, we show many interesting properties such as existence of a universal purely inseparable cover, transitivity of $k$-rational points under $G(k)$ as well as the existence of a smooth parabolic in the stabilizer of every such point. However, these properties fail if $k$ is imperfect and we demonstrate this with illuminating examples.