Specialization of Néron-Severi groups in positive characteristic
Emiliano Ambrosi
TL;DR
The paper proves that in positive characteristic, a smooth proper family f:Y→X exhibits many fibres with the same geometric Néron–Severi rank as the generic fibre, extending known characteristic-zero results. The strategy replaces Hodge-theoretic arguments with crystalline cohomology and the variational Tate conjecture, supported by a novel transfer to overconvergent F-isocrystals and p-adic comparison techniques, augmented by spreading-out to finite fields and independence arguments. A central technical advance is a crystalline–ℓ-adic comparison (Theorem ltocrys) ensuring that ℓ-adic Galois-genericity translates into the crystalline context, enabling NS-genericity results, hyperplane-section corollaries, and uniform Brauer-group bounds in families. The work thus yields new instances of the Tate conjecture for divisors in positive characteristic, informs the behavior of hyperplane sections, and supports uniform boundedness conjectures for Brauer groups in families, with broad implications for families of projective varieties and cyclic covers in characteristic p. Overall, the paper develops a cohesive p-adic-analytic–algebraic framework to study how algebraic cycles deform in families over imperfect fields of characteristic p and demonstrates practical geometric consequences of NS-generic phenomena.
Abstract
Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper is that there are ``lots of" closed points $x\in X$ such that the fibre of $f$ at $x$ has the same geometric Picard rank as the generic fibre. If $X$ is a curve we show, under a minimal technical assumption, that this is true for all but finitely many $k$-rational points. In characteristic zero, these results have been proved by André (existence) and Cadoret-Tamagawa (finiteness) using Hodge theoretic methods. To extend the argument in positive characteristic we use the variational Tate conjecture in crystalline cohomology, the comparison between various $p$-adic cohomology theories and independence techniques. The result has applications to the Tate conjecture for divisors, uniform boundedness of Brauer groups, proper families of projective varieties and to the study of families of hyperplane sections of smooth projective varieties.