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Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed immersion. Denote by $\mathcal{F}$ the sheaf $\mathcal{O}_X(-D)$. Suppose, $\mathcal{F}$ and $i^*\mathcal{F}$ are $d$-regular (in the sense of Castelnuovo-Mumford regularity). Note that this mean $\mathcal{F}(d)$ and $i^*(\mathcal{F}(d))$ are generated by global sections. Assume that $C$ is an irreducible component in the support of $D$. Is it then true that the natural morphism from $H^0(\mathcal{F}(d))$ to $H^0(i^*(\mathcal{F}(d)))$ is surjective?

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What else do we know about $C$? For instance, do we know $O_X(C) \cong O_X(n)$. –  Karl Schwede Feb 3 '14 at 16:33
@Schwede: No. You may assume $C$ is a complete intersection curve in $\mathbb{P}^3$ (not $X$). –  user43198 Feb 3 '14 at 16:39
Ok, then what you want is equivalent to asking for the vanishing $$H^1(X, O_X(-D-C+dH) )$$ where $H$ is the hyperplane class on $X$ (this is because the next term $H^1(X, O_X(-D+dH))$ definitely vanishes by your regularity assumption). I don't see why the first $H^1$ should vanish unfortunately. –  Karl Schwede Feb 3 '14 at 17:02

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