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suppose $C$ is a smooth projective curve. $L$ is a line bundle with a non zero global section $s$. Suppose $L_0$ is another line bundle but of degree zero.

Assume $dimH^0(L)=dimH^0(L\otimes L_0)$. Does $s$ correspond to a canonical section of $L\otimes L_0$ ?

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No reason for that. If $L$ and $L_0$ are generic (given their degrees), then the equality of dimensions will hold. But there is no reason to have a natural isomorphism of spaces of sections.

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