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Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow \mathbb{P}^1$ be the corresponding 2 to 1 covering ramified in $2g+2$ points.

Let $L$ be a line bundle on $C$ such that either $L^{\otimes 2} = \mathcal{O}_C$ or $L^{\otimes 2} = \mathcal{\omega}_C^{\otimes 2}$. Can we find a line bundle $A$ on $\mathbb{P}^1$ such that $f^{*}A = L$.

Any suggestions or references will be most helpful.

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    $\begingroup$ If and only if $L=\mathcal{O}_C$ (first case) or $L=\omega _C$ (second case). Are you aware that $\operatorname{Pic}(\mathbb{P}^1)=\mathbb{Z} $? $\endgroup$
    – abx
    Commented Mar 22, 2018 at 18:09

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