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 CastelnuovoMumford 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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