  Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of
Hilbert's irreducibility theorem holds for families of geometrically mordellic
varieties (for instance, families of hyperbolic curves). As an application we
prove that, assuming Bombieri-Lang, there are no polynomial bijections
$\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}$.
