  Andr\'{e} and Maulik--Poonen proved that for any smooth proper family $X\to
B$ of varieties over an algebraically closed field of characteristic $0$, there
is a closed fiber whose N\'{e}ron-Severi group has the same rank as that of the
N\'{e}ron-Severi group of the geometric generic fiber. We prove the analogous
statement over algebraically closed fields of characteristic $p>0$ which are
not isomorphic to $\overline{\mathbb{F}}_p$. Furthermore, we prove that for any
algebraically closed field $k$ of characteristic $p>0$ and smooth proper family
$X\to B$ of $k$-varieties, there exists a dense open subvariety $U\subseteq B$
and integer $N$ such that for each map $x:\mathrm{Spec}\, k[[t]]\to U$, the
$p$-torsion in the cokernel of the specialization map from the N\'{e}ron-Severi
group of the pullback of $X$ to the geometric generic fiber of $x$ to the
N\'{e}ron-Severi group of the pullback of $X$ to the special fiber of $x$ is
killed by $p^N$. Finally, we prove that for a curve $C$ over $k$ and family
$\mathscr{X}\to\mathscr{B}$ of smooth $C$-schemes, there exists a dense Zariski
open $U\subseteq \mathscr{B}$ such that for a local uniformizer $t$ at any
closed point of $C$, the rank of the N\'{e}ron-Severi group jumps only on a
$t$-adic nowhere dense set $t$. The crystalline Lefschetz $(1,1)$ theorem of
Morrow is a key ingredient in the proofs.
