The Stacks project

Proof. Taking $h$-torsion in the exact sequence of Lemma 55.9.5 we obtain the exactness of $0 \to \mathop{\mathrm{Pic}}\nolimits (X)[h] \to \mathop{\mathrm{Pic}}\nolimits (C)[h]$ because $h$ is prime to $d$. Using the map of Lemma 55.13.2 we get a map $\mathop{\mathrm{Pic}}\nolimits (C)[h] \to \mathop{\mathrm{Pic}}\nolimits (T)[h]$ which annihilates elements of $\mathop{\mathrm{Pic}}\nolimits (X)[h]$. Conversely, if $\xi \in \mathop{\mathrm{Pic}}\nolimits (C)[h]$ maps to zero in $\mathop{\mathrm{Pic}}\nolimits (T)[h]$, then we can find an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ with $\deg (\mathcal{L}|_{C_ i}) = 0$ for all $i$ whose restriction to $C$ is $\xi $. Then $\mathcal{L}^{\otimes h}$ is $d$-torsion by Lemma 55.13.1. Let $d'$ be an integer such that $dd' \equiv 1 \bmod h$. Such an integer exists because $h$ and $d$ are coprime. Then $\mathcal{L}^{\otimes dd'}$ is an $h$-torsion invertible sheaf on $X$ whose restriction to $C$ is $\xi $. $\square$