  In this article we prove a semistable version of the variational Tate
conjecture for divisors in crystalline cohomology, stating that a rational
(logarithmic) line bundle on the special fibre of a semistable scheme over $k
[\![ t ]\!]$ lifts to the total space if and only if its first Chern class
does. The proof is elementary, using standard properties of the logarithmic de
Rham-Witt complex. As a corollary, we deduce similar algebraicity lifting
results for cohomology classes on varieties over global function fields.
Finally, we give a counter example to show that the variational Tate conjecture
for divisors cannot hold with $\mathbb{Q}_p$-coefficients.
