Specialization of Néron-Severi groups in positive characteristic

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a reduced scheme $X$, smooth, geometrically connected, separated 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 that this is true for all but finitely many $k$-rational points. In characteristic zero, these results have been proved by Andre (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 different $p$-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.