The classical Noether-Lefschetz theorem asserts the following: Over the complex numbers, a very general surface $S\subset \mathbb{P}^3$ has Picard number 1 (that is, $Pic(S)\simeq \mathbb Z$), provided that $\mbox{deg }S\ge 4$.

Over finite fields (or even countable fields), the corresponding statement does not make much sense, because here `very general' refers to the surface being chosen outside a countable union of closed proper subsets of the parameter space of surfaces. Still, I think makes sense to ask:

What evidence is there for the Noether-Lefschetz statement over countable fields? I.e., is there a sense in which the set of surfaces with low Picard number constitutes a 'large' proportion of the surfaces in $\mathbb{P}^3$ e.g., over finite fields with many elements?

mightbe odd, so you could refine the question.) – M P Apr 26 '13 at 9:09