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Hi, everyone, I want to ask a question about line bundle.

Let $X$ be a smooth curve over a algebraically closed field $k$, and $f:Y \longrightarrow X$ a Galois finite etale covering with Galois group $G$ and degree $n$. Suppose that $L$ is a line bundle on $X$.

Dose there exist a line bundle $M$ on $Y$ such that $M^{\otimes n}=f^{*}L$?


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Let $C/k$ be a smooth projective curve. Let $U/C$ be a line bundle. If $n|{\rm deg}(U)$ then there exists a line bundle $R$ such that $R^{\otimes n}=U$. This follows from the fact that multiplication by $n$ is an isogeny on the ${\rm Pic}_0(C)$ and thus the tensoring by $n$ morphism ${\rm Pic}_{{\rm deg}(U)/n}\to {\rm Pic}_{{\rm deg}(U)}$ is surjective. This applies to your set-up, with $Y=C$ and $U=f^*L$. See also… – Damian Rössler Nov 26 '12 at 12:27
Why did you make this a comment and not an answer? – Maarten Derickx Nov 28 '12 at 23:43

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