Specialization of N\'eron-Severi groups in positive characteristic

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 $f:Y\to X$. 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 Andr'e (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.