N\'eron-Severi groups under specialization

TL;DR

This paper presents a new p-adic approach to understanding how the Picard number behaves in families of varieties, showing that the locus of jumping Picard numbers is p-adically sparse, with implications for higher codimension cycles.

Contribution

It offers a novel p-adic proof of André's theorem and extends results to higher codimension cycles under a p-adic variational Hodge conjecture.

Findings

The locus where the Picard number jumps is nowhere p-adically dense.

The p-adic Lefschetz (1,1) theorem is effectively used in the proof.

Analogous results are established for higher codimension cycles.

Abstract

Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to 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. We give a completely different approach to Andr\'e's theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is nowhere p-adically dense. Our proof uses the ``p-adic Lefschetz (1,1) theorem'' of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.