Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 167 / 3 / ii / Solution

A flat algebraic-group torsor $X\to Y$ is a faithfully flat morphism with a right $H$-action for whichis an isomorphism.

The orbit $G\ell$ is a locally closed subvariety of $\mathbb{P}(V)$ by the orbit theorem for algebraic-group actions. The fibers of the orbit map $G\to G\ell$ are precisely the right cosets of $H={\mathrm{Stab}}_G(\ell )$, and the displayed action map is therefore an isomorphism. The theorem on quotients of affine algebraic groups by closed subgroups says that $G/H$ exists and $G\to G/H$ is faithfully flat; the induced map $G/H\to G\ell$ is an isomorphism. Hence the orbit map is a flat $H$-torsor.