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The set of maximal tori defined over $k$ is finite: if $T$ is one of them, they correspond to the $k$-rational points of the $k$-variety $G/\text{(normalizer of }T)$. So their union is not dense, unle …

In what follows, we only need $G_1$ to be normal in $G$, as in the last question. Put $Y=X/G$, and $Z=X/G_1$: $Y$ is a variety by assumption, and both $X$ and $Z$ make sense as sheaves on the étale si …

Yes: the following examplpe is quite different from the example by Ariyan, although the generic and closed fibers are the same as his. Start with the constant group scheme $A:=(\mathbb{Z}/2\mathbb{Z …

$G/H$ is affine. More generally, assume $G$ is an affine algebraic group over a field $k$ (the affine condition is needed, although the OP does not mention it) and $H<N\lhd G$ be $k$-subgroups (with $ …

This holds for any homomorphism $f: G\to H$ with $G$ of multiplicative type and of finite type, and $H$ separated and finitely presented. Here I assume that by "finite flat" you mean "finite locally f …

Let $b:T\times T\times T\to T$ be the map in question. We can view it as a morphism of functors (actually fpqc sheaves) on $k$-schemes: $$\begin{array}{rcl} T\times T & \longrightarrow & \operatorname …

The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology. No ground field is needed: $Y$ could be any …

The claim is false. Over a field of characteristic $p>0$, take for $G$ the semidirect product $\alpha_p\rtimes\mathbb{G}_\mathrm{m}$ where $\mathbb{G}_\mathrm{m}$ acts on $\alpha_p$ by scaling. Then $ …

If $\mathrm{char}(k)=p>0$ and $G$ is a $k$-group scheme of finite type, the kernel of the relative frobenius $F_{G/k}:G\to G^{(p)}$ is a finite connected $k$-group scheme. It has the same Lie algebra …

For every $\mathbb{R}$-scheme $X$ of finite type, $\pi_0(X(\mathbb{R}))$ is finite. This follows e.g. from Theorem 2.3.6 in Bochnak, Coste, Roy, Real Algebraic Geometry (basic structure theorem for …

No, it does not follow that $G_\mathrm{red}$ is geometrically reduced (thus the answer to the title question is yes). Let $p=\mathrm{char}(k)>0$ and let $H=\mathbb{G}_a \rtimes \mathbb{G}_m$ be the gr …

If $k$ is a field, the Hilbert functor of closed subschemes of a $k$-scheme of finite type $X$ is representable if $X$ is projective (or more generally quasiprojective if you restrict to projective su …

No. For one thing, it would imply that every group subsheaf $K$ of an (abelian) algebraic space in groups $H$ is an algebraic space (just take $G=H/K$). Now let me give a "concrete" example. For an …

Take $p=3$ and $C\subset \mathbb{P}^2$ with equation $y^2 z=x^3 - t z^3$ where $t\in K$ is not a cube. Then $C$ is regular but, putting $L:=K(t^{1/3})$, $C_L$ is isomorphic to the usual cuspidal cubic …