The classical Noether–Lefschetz theorem is the following:
Theorem. Let be a very general smooth surface of degree . Then the natural map is an isomorphism.
If is a smooth proper family over some base (usually of finite type over a field), then a property holds for a very general if there exists a countable intersection of nonempty Zariski opens such that holds for for all .
In general, Hilbert scheme arguments show that the locus where the Picard rank is ‘bigger than expected’ is a countable union of closed subvarieties of (the Noether–Lefschetz loci), but it could be the case that this actually happens everywhere (i.e. ). The hard part of the Noether–Lefschetz theorem is that the jumping loci are strict subvarieties of the full space of degree hypersurfaces.
If is a family of varieties over an uncountable field , then there always exists a very general member with . But over countable fields, very general elements might not exist, because it is possible that even when .
The following interesting phenomenon was brought to my attention by Daniel Bragg (if I recall correctly):
Example. Let (the algebraic closure of the field of elements, but the bar is not so visible in MathJax), let (or some scheme covering it if that makes you happier) with universal family of elliptic curves, and let be the family of product abelian surfaces . Then the locus
is exactly the set of -points (so it misses only the generic point).
Indeed, , and every elliptic curve over has . But the generic elliptic curve only has .
We see that the Noether–Lefschetz loci might cover all -points without covering , even in very natural situations.