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Suppose I have a stable, reducible, nodal curve with two components $C' \cup C''=C$ and two line bundles $L$ and $M$ on $C$. Suppose furthermore that $L^{\otimes r}=M$. What can I say about the restrictions $L'$ and $L''$ of $L$ to $C'$ and $C''$. Do we have $L_{|C'}^{\otimes r}=M_{|C'}$? I guess that the two bundles $L'$ and $L''$ should have some condition on the attaching point. (residue sum up to zero?)

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Both isomorphism and tensor-product are compatible with arbitrary pullback, thus, in particular, with restriction to irreducible components. –  Jason Starr Aug 29 '13 at 12:38
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