Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero section to $\pi$. If I understand correctly the morphism $\pi^*:\mathrm{Pic}(X) \to \mathrm{Pic}(\mathbb{L})$ is an isomorphism. Then, what is the preimage of the invertible sheaf $\mathcal{O}_{\mathbb{L}}(0_\pi)$ on $\mathbb{L}$, under this morphism?

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    $\begingroup$ The inverse isomorphism is $s^*$, where $s:X\to \mathbb{L}$ is the zero section morphism. Thus, depending on your sign convention (which you have not specified), the pullback $s^*\mathcal{O}(\underline{\text{Image}(s)})$ is either $L$ of the dual of $L$. $\endgroup$ – Jason Starr Mar 8 '18 at 13:45

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