Very special algebraic groups

Michel Brion,Emmanuel Peyre

doi:10.5802/crmath.86

Abstract

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.

Highlights

Consider a smooth algebraic group G over a field k, and a G-variety X
We say that a smooth algebraic group G over a field k is very special if for any field extension K /k, every GK -homogeneous K -variety has a K -rational point
We show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions

Summary

Introduction

Consider a smooth algebraic group G over a field k, and a G-variety X. [11, §2.7]) motivates the consideration of those smooth algebraic groups for which all rational quotients have rational sections These are described as follows: Theorem 2. Every split solvable linear algebraic group G satisfies a much stronger condition: for any field extension K /k, every GK -homogeneous variety is rational (as follows from [14, Thm. 5]). One may consider algebraic groups G that are possibly non-smooth, and require that for any field extension K /k, every GK -homogeneous K -scheme has a K -rational point (where a scheme X equipped with an action a of G is said to be homogeneous if the graph morphism id ×a : G × X → X × X is faithfully flat). Every homogeneous space is smooth and quasi-projective; so is every homogeneous variety

Proof of Proposition 1

Proof of Theorem 2: first steps

Very special tori

Completion of the proof of Theorem 2