# Highlights in Linear Algebraic Groups 10: G/B is Projective

In Highlights 9 of this series, we showed that for an algebraic group $G$ and a closed subgroup $H\subseteq G$, we can always choose a representation $G\to\rm{GL}(V)$ with a line $L\subseteq V$ whose stabiliser is $H$. In turn, this allows us to identify the quotient $G/H$ with the orbit of the class $[L]$ in the projective space $\mathbf{P}(V)$, which satisfies the universal property for quotients, thereby giving us a sensible variety structure on $G/H$.

In this post, we specialise to the case of a Borel subgroup $B\leq G$; that is $B$ is maximal amongst the connected solvable groups. Such a subgroup is necessarily closed!

The fact that will allow us to study Borel subgroups is the fixed point theorem: a connected solvable group that acts on a nonempty complete variety has a fixed point. By choosing a representation $G\to \rm{GL}(V)$ with a line $L\subseteq V$ whose stabiliser is $B$, we get identify $G/B$ with a quasiprojective variety. However, in this case $G/B$ is actually projective. Here is a short sketch:
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