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There were quite a few different questions, so forgive me if my answer is somewhat fragmented. The Néron-Severi group $NS(X)$ (divisors modulo algebraic equivalence) is finitely generated over any fi …

My guess is that this is true, however there is basically no hope in proving this at all in general. The most relevant conjecture here is Manin's conjecture [1]. As an example, take $S$ to be a smoot …

How about $$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-2} \quad ?$$ This is a fairly well-known example which seems to satisfy your criteria.

It is not too difficult to see that any automorphism of a smooth quadric hypersurface $$X : Q(x) = 0,$$ over a field $k$ must be a projective automorphism (see for instance the argument I give in Auto …

No: Consider the elliptic curve $E: y^2 = x^3 + x$ defined over $\mathbb{Q}$. Then the isogeny $y \mapsto iy$ and $x \mapsto -x$ is defined over $\mathbb{Q}(i)$ but obviously not over $\mathbb{Q}$. I …

Let $X$ be an integral regular scheme which is proper over $\mathbb{Z}_p$. Assume that the special fibre $X_{\mathbb{F}_p}$ is irreducible and let $k$ be the algebraic closure of $\mathbb{F}_p$ in the …

I hope nobody minds me answering this (very old) question. Me and my collaborators (Barinder Banwait and Francesc Fité) succeeded in completely answering this question in the paper: Del Pezzo surfac …

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 …

As explained in the comments, there is no such variety. This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the bo …

No: this fails already when $X=E$ is an elliptic curve and $k=\mathbb{Q}$. This would imply that every element of $H^1(\mathbb{Q},E)$ has order at most $d$, and I'm pretty sure that this cohomology gr …

(Upgrading comments to answer.) Counting the number of $D$ for which the equation has a rational solution is a fairly classical problem. This is asymptotic to $c_nD/(\log D)^{1/2}$ for some $c_n > 0$. …

Surely this behaviour can never happen, but it will be near impossible to prove this. Conjectures of Manin and others predict that there is an asymptotic formula for these functions in many cases, and …

This is a brief answer; possibly others have different opinions about this. Question 1: The Langlands conjectures gives a correspondence between Galois representations and automorphic forms. So a ver …

I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to …

Suppose that $C \subset X$ is a smooth projective curve of genus $1$ embedded in a Brauer-Severi surface over a field $k$. We have $C^2 = 9$ since this holds after passing to the algebraic closure, wh …