Very special algebraic groups

Brion, Michel; Peyre, Emmanuel

Comptes Rendus. Mathématique, Tome 358 (2020) no. 6, pp. 713-719.

Nous disons qu’un groupe algébrique lisse $G$ sur un corps $k$ est très spécial si pour toute extension de corps $K/k$, toute $K$-variété homogène sous $G_K$ a un point $K$-rationnel. On sait que tout groupe linéaire résoluble scindé est très spécial. Dans cette note, nous obtenons la réciproque et nous discutons ses relations avec la classification birationnelle des actions de groupes algébriques.

We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.

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