Lemma 55.13.3. In Situation 55.9.3 let $d = \gcd (m_1, \ldots , m_ n)$ and let $T$ be the numerical type associated to $X$. Let $h \geq 1$ be an integer prime to $d$. There exists an exact sequence

$0 \to \mathop{\mathrm{Pic}}\nolimits (X)[h] \to \mathop{\mathrm{Pic}}\nolimits (C)[h] \to \mathop{\mathrm{Pic}}\nolimits (T)[h]$

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 Lemma 55.13.1 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 $\xi$. $\square$

Comment #7999 by Fabio Bernasconi on

'Using the map Lemma 0CAB we get a map Pic(C)[h]â†’Pic(T)[h]' Should it refer to tag/0CAC where you construct the map Pic(C) \to Pic(T)?

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