Fixed Point Theorem for F-Split Solvable Groups
Let \(H\) be a connected solvable \(F\)-group that is \(F\)-split. Let \(X\) be a projective \(F\)-variety with an \(F\)-defined action of \(H\). If \(X(F)\neq\emptyset\), then there exists a point \(x\in X(F)\) fixed by all elements of \(H\).
Proof
Choose an \(F\)-split composition series \[ \{e\}=H_0\subset H_1\subset \cdots \subset H_n=H \] whose successive quotients are \(F\)-isomorphic to \(\mathbb{G}_a\) or \(\mathbb{G}_m\). Starting from an \(F\)-point \(x_0\in X(F)\), construct inductively an \(F\)-point \(x_i\) fixed by \(H_i\).
If \(H_i\) already fixes \(x_{i-1}\), set \(x_i=x_{i-1}\). Otherwise the orbit map of \(H_i/H_{i-1}\) through \(x_{i-1}\) gives an \(F\)-morphism from \(\mathbb{G}_a\) or \(\mathbb{G}_m\) into \(X\). Since \(X\) is projective, this morphism extends to \(\mathbb{P}^1\). The point at infinity is \(F\)-rational and fixed by the quotient action, giving \(x_i\). At the end, \(x_n\) is fixed by \(H\).