An interesting Noether–Lefschetz phenomenon

The classical Noether–Lefschetz theorem is the following:

Theorem. Let $X \subseteq \mathbf P^3_{\mathbf C}$ be a very general smooth surface of degree $d \geq 4$ . Then the natural map $\Pic(\mathbf P^3) \to \Pic(X)$ is an isomorphism.

If $\mathscr X \to S$ is a smooth proper family over some base $S$ (usually of finite type over a field), then a property $\mathcal P$ holds for a very general $X = \mathscr X_s$ if there exists a countable intersection $U = \bigcap_i U_i \subseteq S$ of nonempty Zariski opens $U_i$ such that $\mathcal P$ holds for $X_s$ for all $s \in U$ .

In general, Hilbert scheme arguments show that the locus where the Picard rank is ‘bigger than expected’ is a countable union of closed subvarieties $Z_i$ of $S$ (the Noether–Lefschetz loci), but it could be the case that this actually happens everywhere (i.e. $U = \varnothing$ ). The hard part of the Noether–Lefschetz theorem is that the jumping loci $Z_i$ are strict subvarieties of the full space of degree $d$ hypersurfaces.

If $\mathscr X \to S$ is a family of varieties over an uncountable field $k$ , then there always exists a very general member $\mathscr X_s$ with $s \in S(k)$ . But over countable fields, very general elements might not exist, because it is possible that $\bigcup Z_i(k) = S(k)$ even when $\bigcup Z_i \neq S$ .

The following interesting phenomenon was brought to my attention by Daniel Bragg (if I recall correctly):

Example. Let $k = \bar{\mathbf F}_p$ (the algebraic closure of the field of $p$ elements, but the bar is not so visible in MathJax), let $S = \mathcal A_1 = \mathcal M_{1,1}$ (or some scheme covering it if that makes you happier) with universal family $\mathscr E \to S$ of elliptic curves, and let $\mathscr X = \mathscr E \times_S \mathscr E$ be the family of product abelian surfaces $E \times E$ . Then the locus

$NL(S) = \left\{s \in S\ \big| \ \operatorname{rk} \Pic(\mathscr X_s) > 3\right\}$

is exactly the set of $k$ -points (so it misses only the generic point).

Indeed, $\Pic(E \times E) \cong \Pic(E) \times \Pic(E) \times \End(E)$ , and every elliptic curve $E$ over $k$ has $\operatorname{rk} \End(E) \geq 2$ . But the generic elliptic curve only has $\End(E) = \mathbf Z$ . $\qedsymbol$