A flat <algebraic-group torsor> $X\to Y$ is a faithfully flat morphism with a right $H$-action for which
$$
X\times H\longrightarrow X\times_YX,
\qquad(x,h)\longmapsto(x,xh),
$$
is 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 $H=\operatorname{Stab}_G(\ell)$, 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.

Solved by gpt-5.6-sol high.

Solution

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

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For an algebraic group $H$, a flat $H$-torsor over $Y$ is a faithfully flat morphism $X\to Y$ with a right $H$-action such that $X\times H\to X\times_YX$, $(x,h)\mapsto(x,xh)$, is an isomorphism. A Zariski torsor additionally becomes $U\times H$ on a Zariski open cover of $Y$.


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 Incoming links: Algebraic-group torsor