The Stacks project

Proof. By Lemma 108.8.5 we know that $\mathcal{P}\! \mathit{ic}_{X/B} \to B$ is smooth. The morphism $\mathcal{P}\! \mathit{ic}_{X/B} \to \mathrm{Pic}_{X/B}$ is surjective and smooth by combining Lemma 108.9.1 with Morphisms of Stacks, Lemma 101.33.8. Thus if $U$ is a scheme and $U \to \mathcal{P}\! \mathit{ic}_{X/B}$ is surjective and smooth, then $U \to \mathrm{Pic}_{X/B}$ is surjective and smooth and $U \to B$ is surjective and smooth (because these properties are preserved by composition). Thus $\mathrm{Pic}_{X/B} \to B$ is smooth for example by Descent on Spaces, Lemma 74.8.3. $\square$