The Stacks project

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

  1. restricts to the trivial invertible module on $C$, and

  2. 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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