Borel fixed point theorem for complete schemes

Given: The Axiom of Choice, a field $k$, a smooth connected solvable affine algebraic group $G$ over $k$, and a nonempty complete finite-type $k$-scheme $X$ with an action of $G$.

Base change along $k\to k^{\mathrm{a}}$ preserves completeness, nonemptiness and finite type, and the action base changes; a fixed point over $k^{\mathrm{a}}$ is exactly a point of $X(k^{\mathrm{a}})$ fixed by $G(k^{\mathrm{a}})$. Smoothness survives field extension; connectedness does so for group schemes by geometric connectedness, and solvability does so because the derived-subgroup construction commutes with field extension. Completeness here means properness, including for nonreduced $X$. (Milne A.75 and A.76, printed p. 587; Connected finite-type groups are geometrically connected, Properties of the derived subgroup of an algebraic group, Proper morphisms)

If $\dim G=0$, then $G=1$: a smooth connected finite-type group scheme of dimension zero is a single reduced point. A nonempty finite-type scheme over an algebraically closed field has a $k$-point: take a nonempty affine chart $\mathrm{Spec}B$, choose a maximal ideal in its nonzero finitely generated algebra by AC, and apply the weak Nullstellensatz to its preimage under a polynomial-ring surjection onto $B$. (Smooth morphism of schemes, Connected finite-type groups are geometrically connected, In a nonzero commutative ring, every proper ideal is contained in a maximal ideal, Over an algebraically closed field, every maximal ideal is an evaluation ideal)

Assume AC. If $G$ is smooth, connected and solvable with $G\mathrm{\ne }1$, then $DG$ is a smooth connected closed characteristic normal subgroup scheme with $DG\mathrm{\ne }G$, so $\dim DG<\dim G$. (Properties of the derived subgroup of an algebraic group)

Assume AC. Let $G$ be smooth over an algebraically closed field acting on a separated finite-type scheme $X$, let $H\subseteq G$ be a smooth closed normal subgroup scheme and $y\in X(k)$ fixed by $H(k)$. Then the closure $Z$ of the $G$-orbit of $y$ is $G$-stable and fixed pointwise by $H(k)$; and for every $x\in Z(k)$ fixed by $H(k)$, the stabilizer $G_x$ contains $H$. (Fixed loci are closed and a normal subgroup fixing a point fixes the orbit closure)

Assume AC. For a smooth group $G$ acting on a separated finite-type scheme $X$, the orbit map $G\to G\cdot x$ is faithfully flat, the orbit of a $k$-point of minimal dimension among the orbits in a $G$-stable closed subset is closed, and for an orbit of minimal dimension the orbit map exhibits the orbit as the coset space $G\mathrm{/}{G}_x$, a separated finite-type scheme. (Smooth orbits are locally closed and their orbit maps are faithfully flat over every field, Fibre dimension and orbit dimension add to the dimension of the group, A faithfully flat orbit map represents the coset quotient sheaf, Homogeneous spaces of smooth affine groups are separated schemes)

Assume AC. If $N$ is a closed normal subgroup scheme of the affine group $G$, the quotient $G\mathrm{/}N$ is affine; if $G_x$ is a closed subgroup containing $DG$, then $G_x$ is normal in $G$. A reduced connected complete affine finite-type $k$-scheme over algebraically closed $k$ is a single reduced point: properness makes its coordinate algebra finite-dimensional, reducedness makes it a product of finite field extensions of $k$, and connectedness leaves one factor, equal to $k$. The reducedness condition excludes infinitesimal counterexamples such as ${\alpha }_p$. (Quotients of affine group schemes by normal subgroup schemes are affine, Properties of the derived subgroup of an algebraic group, Morphisms from complete connected schemes to affine schemes are constant)