Griffiths and Harris, On the Noether-Lefschetz Theorem and Some Remarks on Codimension-Two Cycles, Math. Ann. 271, 31-51 (1985), states
[...] look at the restriction $$r_1 : \operatorname{Pic}({\widetilde{S}}) \to \operatorname{Pic}(\widetilde{C})$$
[...]
We make the following observations:
[...]
ii) $r_1$ is injective, since the curve $\widetilde{C}$ is a generically chosen hyperplane section of the non-ruled surface $\widetilde{S}$.
on page 36. (Paper available here: http://publications.ias.edu/sites/default/files/noether.pdf)
Question: Why is this true?