  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 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.
