Projective homogeneous varieties of Picard rank one in small characteristic
Résumé
We extend to characteristic $2$ and $3$ the classification of projective homogeneous varieties of Picard group $\Z$, corresponding to parabolic subgroups with maximal reduced subgroup. In all types, except for $G_2$ in characteristic $2$, the latter are all obtained as product of a maximal reduced parabolic with the kernel of a purely inseparable isogeny. For the $G_2$ case, we exhibit an explicit counterexample and show it is the only one, thus completing the classification. We then construct new examples of projective homogeneous varieties of Picard rank two.
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