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As phrased, this problem looks just as difficult as determining whether a given element of $\mathbb{Q}_p$ lies in $\mathbb{Q}$. There are real numbers which are unknown to be rational (e.g. $\pi + e$) …

It is the Fermat cubic surface over $\mathbb{F}_4$ or (if you prefer $\mathbb{F}_p$) over $\mathbb{F}_7$. There are quite a few papers on this topic. Firstly in [1] Swinnerton-Dyer showed (amongst oth …

The torus $T$ may have non-trivial Picard group. Therefore this sequence is not exact in general. One may consider for example $T: x^2 + y^2 = 1$ over $\mathbb{Q}$. The closure of this is a conic, and …

It is not clear from the question whether you want to determine whether $X$ admits a conic bundle structure, or whether it is birational to a conic bundle surface. I will focus on the former. If you a …

This is an open problem; I personally suspect it cannot happen. More generally let $X$ be a smooth projective variety over a field $k$ which is finitely generated over $\mathbb{Q}$. Then Skorobogatov …

Probably the easiest way to prove this is via flatness. The closure $\bar{x}$ is integral and dominates $S$, thus is flat over $S$ (see Proposition III.9.7 in Hartshorne). The dimension of the fibres …

What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given. Note that …

Let $k$ be a field with algebraic closure $\bar{k}$. Recall that a gerbe over $k$ is an algebraic stack $\mathcal{G}$ over $k$ such that the groupoid $\mathcal{G}(\bar{k})$ is connected. We say that $ …

This is not possible in any meaningful way. In fact the variety you describe defines a smooth cubic threefold $X$ in $\mathbb{P}^4$. By a famous theorem of Clemens and Griffiths these are not even rat …

Let $\varphi:X \to Y$ be a smooth morphism of schemes. There is a well-known local structure theorem: Zariski locally $\varphi$ is given by the composition of an etale morphism and an affine space (s …

Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be $$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}. …

Is the Picard group of a Fano variety always finitely generted and torsion free? This is well known over fields of characteristic 0, so the question is about the case of positive characteristic. Note …

Firstly $\mathbb{G}_a$ and $\mathbb{G}_m$ are definitely not isomorphic as group schemes even in characterstic $0$, as the exponential is not an algebraic map. But there is a foundational paper on the …

Let $k$ be a global field and $C$ a smooth projective curve over $k$ which is not isotrivial. Then there is a well-known trichotomy: If $g(C) = 0$ and $C(k) \neq \emptyset$, then $C \cong \mathbb{P}^ …

Let $X$ be a smooth projective rationally connected variety over $\mathbb{C}((t))$ and $R= \mathbb{C}[[t]].$ Does there exist a proper regular scheme $\mathcal{X} \to \mathrm{Spec}(R)$ whose special f …