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I think there is some confusion in your question; I agree with René that probably the correct analogue of rational point in general is something like a geometrically integral subscheme. I will howeve …

Let $k$ be a field of characteristic not equal to $2$ and let $$C: y^2 = f(x)$$ be an elliptic curve over $k$. Then a quadratic twist of $k$ is a curve of the form $$C_d: dy^2 = f(x)$$ for $d \in k^*$ …

Let $f: X \to S$ be a morphism of algebraic spaces, where $S$ is a scheme. If $f$ is separated and étale then Knutson's criterion says that $X$ is actually a scheme. I have a some closely related que …

I've heard in informal conversations before the claim that: "a generic singular hypersurface has a single singularity of type $\mathbf{A}_1$". What is the precise statement of this result? Where can …

Here is a method which should give you what you want, but I leave the details to you as I don't have time to work them out now. Without loss of generality we may work over an algebraically closed fie …

For the cases which you are interested in there appears to be only partial results available in the literature. Namely, as special cases of more general results. For quartic threefolds and cubic five …

Yes to your first question. See the paper Koitabashi - Automorphism groups of generic rational surfaces. As for your second question, whilst Kotabashi uses some standard techniques on the geometry …

The degree map gives a short exact sequence $$ 0 \to J(C) \to Pic(C) \to \mathbb{Z} \to 0,$$ where $J(C)$ denotes the Jacobian of $C$. This is an abelian variety of dimension $g$, where $g$ is the gen …

Here is another way to see why this is true (though as already noted you need $s >3$). Note that if $X$ is a toric variety with respect to an algebraic torus $T$, then by definition $T$ acts faithful …

Im not sure if this counts as a full answer, but it is a nice example which will hopefully shed light on some of your questions. The canonical bundle $\omega_X$ of an Enriques surface $X$ satisfies $ …

Let $G$ be a group scheme over a scheme $X$ with centre $Z(G)$, automorphism group $\mathrm{Aut}(G)$ and outer automorphism group $\mathrm{Out}(G)$ (viewed as group schemes on $X$). If $G$ is f …

The answer to 2. is yes. A sketch of a proof is as follows: $G$, being finite étale, is étale locally on $X$ isomorphic to a constant finite group scheme. Therefore, by a standard descent argument, i …

Yes. This holds for any cubic surface over an algebraically closed field $k$. Let $S$ be such a surface. Let $L'$ be a line. The pencil of hyperplanes containing $L'$ forms a conic bundle $\pi: S \to …

Note that there is some small ambiguity here, as to talk about the reduction of $X/\mathbb{Q}$ modulo a prime $p$, one needs to choose a model $X$. i.e., a scheme $\mathcal{X} \to \mathbb{Z}$ whose ge …

The answer to question 1 is yes. Examples almost like what you are after can be found here: Does isomorphic generic fibre imply isomorphic special fibre for smooth morphisms? Though admittedly the exa …