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Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$ and $C_1, C_2$ be integral curves on $X$. Let $n$ be such that the ideal sheaf $\mathcal{I}_{C_1 \cup C_2}(n)$ is globally generated as an $\mathcal{O}_{\mathbb{P}^3}$-module. Assume that $C_1 \cup C_2$ is connected. Under what condition on $C_1$ and $C_2$ is the natural morphism $H^0(X,\mathcal{I}_{C_1 \cup C_2}(n)) \to H^0(C_1, \mathcal{I}_{C_1 \cup C_2} \otimes_{\mathcal{O}_{\mathbb{P}^3}} \mathcal{O}_{C_1}(n))$ surjective? For example if $C_1, C_1 \cup C_2$ are both complete intersection curves in $\mathbb{P}^3$ (not in $X$) can this hold true?

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