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Let $C_1$ and $C_2$ be projective curves in $\mathbb{P}^3$. Assume further that $C_1 \cap C_2$ is finitely many points. Let $\mathcal{F}$ (resp. $\mathcal{G}$) be a sheaf defined over $C_1$ (resp. $C_1 \cup C_2$). Assume that $\mathcal{G}|_{C_1\backslash C_2}=\mathcal{F}|_{C_1\backslash C_2}$. Given any global section $s$ of $\mathcal{F}$ can we ''extend'' this to a global section $s'$ of $\mathcal{G}$ in the sense does there exist a global section $s'$ of $\mathcal{G}$ such that it agrees with $s$ on $C_1\backslash C_2$? If this is false in general, can we impose some condition on the sheaves under which this will be possible?

Chen
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  • Certainly not true in general: let $C_1$ and $C_2$ be lines meeting in a point $p$, and let $F=O_{C_1}$. Then for $G$ we can take any line bundle $L$ on $C_2$, and choose an isomorphism of the fibres at $p$. But now if we choose $s$ to be a nonzero constant function, and choose $L$ to be a bundle with no global section, it's clear that $s$ cannot extend to $C_2$. –  Sep 25 '13 at 12:28

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