NAG

Abstract

This is joint work with Davesh Maulik and Claire Voisin. We prove that given a smooth proper family X --> B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. In fact, we give two proofs, and they show that the locus on the base where the Picard number jumps is "small" in two different senses. The first proof uses Hodge theory and the actions of geometric monodromy groups and Galois groups to show that the locus is small in a sense related to Hilbert irreducibility. The second proof uses the "p-adic Lefschetz (1,1) theorem" of Berthelot and Ogus to show that in a family of varieties with good reduction at p, the locus is nowhere p-adically dense. Finally, we prove analogous statements for cycles of higher codimension, under the assumption of the variational Hodge conjecture or a p-adic analogue conjectured by M. Emerton.