The Stacks project

Proof. Let $T$ be a scheme and let $(\mathit{Sch}/T)_{fppf} \to \mathcal{P}\! \mathit{ic}_{X/B}$ be given by the object $\xi = (T, h, \mathcal{L})$ of $\mathcal{P}\! \mathit{ic}_{X/B}$ over $T$. We have to show that

is representable by a scheme $V$ and that the corresponding morphism $V \to T$ is surjective and smooth. See Algebraic Stacks, Sections 94.6, 94.9, and 94.10. The forgetful functor $\mathcal{P}\! \mathit{ic}_{X/B, \sigma } \to \mathcal{P}\! \mathit{ic}_{X/B}$ is faithful on fibre categories and for $T'/T$ the set of isomorphism classes is the set of isomorphisms

See Algebraic Stacks, Lemma 94.9.2. We know this functor is representable by an affine scheme $U$ of finite presentation over $T$ by Proposition 99.4.3 (applied to $\text{id} : T \to T$ and $\mathcal{O}_ T$ and $\sigma ^*\mathcal{L}$). Working Zariski locally on $T$ we may assume that $\sigma _ T^*\mathcal{L}$ is isomorphic to $\mathcal{O}_ T$ and then we see that our functor is representable by $\mathbf{G}_ m \times T$ over $T$. Hence $U \to T$ Zariski locally on $T$ looks like the projection $\mathbf{G}_ m \times T \to T$ which is indeed smooth and surjective. $\square$