The Stacks project

Proof. We already know that $\mathcal{P}\! \mathit{ic}_{X/B}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ by Examples of Stacks, Lemma 95.16.1. Let us show descent for objects for $\mathcal{P}\! \mathit{ic}_{X/B, \sigma }$. Let $\{ T_ i \to T\} $ be an fppf covering and let $\xi _ i = (T_ i, h_ i, \mathcal{L}_ i, \alpha _ i)$ be an object of $\mathcal{P}\! \mathit{ic}_{X/B, \sigma }$ lying over $T_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. Applying the result for $\mathcal{P}\! \mathit{ic}_{X/B}$ we see that we may assume we have an object $(T, h, \mathcal{L})$ of $\mathcal{P}\! \mathit{ic}_{X/B}$ over $T$ which pulls back to $\xi _ i$ for all $i$. Then we get

Since the maps $\varphi _{ij}$ are compatible with the $\alpha _ i$ we see that $\alpha _ i$ and $\alpha _ j$ pullback to the same map on $T_ i \times _ T T_ j$. By descent of quasi-coherent sheaves (Descent, Proposition 35.5.2, we see that the $\alpha _ i$ are the restriction of a single map $\alpha : \mathcal{O}_ T \to \sigma _ T^*\mathcal{L}$ as desired. We omit the proof of descent for morphisms. $\square$