Lemma 55.13.1. In Situation 55.9.3 let d = \gcd (m_1, \ldots , m_ n). If \mathcal{L} is an invertible \mathcal{O}_ X-module which
restricts to the trivial invertible module on C, and
has degree 0 on each C_ i,
then \mathcal{L}^{\otimes d} \cong \mathcal{O}_ X.
Proof. By Lemma 55.9.5 we have \mathcal{L} \cong \mathcal{O}_ X(\sum a_ i C_ i) for some a_ i \in \mathbf{Z}. The degree of \mathcal{L}|_{C_ j} is \sum _ j a_ j(C_ i \cdot C_ j). In particular (\sum a_ i C_ i \cdot \sum a_ i C_ i) = 0. Hence we see from Lemma 55.9.7 that (a_1, \ldots , a_ n) = q(m_1, \ldots , m_ n) for some q \in \mathbf{Q}. Thus \mathcal{L} = \mathcal{O}_ X(lD) for some l \in \mathbf{Z} where D = \sum (m_ i/d) C_ i is as in Lemma 55.9.8 and we conclude. \square
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