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

We see that the Noether–Lefschetz loci might cover all k-points without covering S, even in very natural situations.

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