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

For $G={\mathbb{G}}_m$ and $H={\mu }_2$, the quotient isIt is a faithfully flat ${\mu }_2$-torsor, including in characteristic two where ${\mu }_2$ is nonreduced. It is not a Zariski torsor: a local section over any nonempty open set would put a square root of the coordinate $t$ in the function field $k(t)$, but $t$ is not a square there.

Take $V=k\oplus k$ with $z\in {\mathbb{G}}_m$ acting by $z\cdot (x,y)=(x,z^{2}y)$, and let $\ell =k(1,1)$. The equality $z\ell =\ell$ holds exactly when $z^2=1$, so the scheme-theoretic stabilizer is ${\mu }_2$.