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Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $C$ be a smooth projectively normal curve in $X$ and $\mathcal{L}$ be a line bundle on $X$ of the form $\mathcal{O}_X(-D)$ for some (reduced) curve $D$. Let $k$ be a positive integer such that $\mathcal{L} \otimes_X \mathcal{O}_C$ is $k$-regular (in the sense of Castelnuovo-Mumford regularity). Does there exist a positive integer $n$ such that the dimension of the kernel of the natural morphism from $H^0(\mathcal{L} \otimes_X \mathcal{O}_C (k)) \otimes H^0(\mathcal{O}_{\mathbb{P}^3}(n))$ to $H^0(\mathcal{L}\otimes_X \mathcal{O}_C(k+n))$ is strictly less than $h^0(\mathcal{O}_{\mathbb{P}^3}(n))$?

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I think so. Take a general genus 4 curve, embedded canonically in $\mathbb{P}^3$ as a $(2,3)$ complete intersection. Take $\mathcal{L}=\mathcal{O}$, $k=1$ and $n=2$. Then $h^0(\mathcal{O}(3))=20$, the multiplication map seems to have Kernel of dimension 4. – IMeasy Nov 25 '13 at 20:37
I have the feeling that you have something different in mind or did not reveal all the conditions. If $\mathcal L$ is arbitrary, this is clearly trivially true. In other words, for any $k\in \mathbb N$ there exist infinitely many $\mathcal L$ such that the desired condition holds for any $n\in \mathbb N$, namely take $\mathcal L=\mathscr O_{\mathbb P^3}(-k-m)|_C$ for any $m\in\mathbb N_{+}$. Then the left term of the tensor product is zero and hence so is the kernel.Requiring that this group is not zero is no help, with a little work you can figure out that it would work for $m=0$ as well.... – Sándor Kovács Nov 26 '13 at 5:56
@Kovacs: I have edited the question. – user43198 Nov 26 '13 at 9:23
@user43198: the above examples still work. – Sándor Kovács Nov 26 '13 at 10:04
@Kovacs: I am sorry. I am a bit confused. The above example is a line bundle of a specific form (pull-back of the dual of an ample line bundle). Does it also hold for any line bundles as in the question? – user43198 Nov 26 '13 at 10:07

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