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Nov 12, 2012 at 10:33 comment added Qing Liu Under you assumption, $\mathcal M$ is supported in a closed finite subset of $X$. There is a surjective homomorphism $O_X\to M$ if and only if for all $x\in X$, $\dim_{\mathbb C}(M/\mathfrak m_x M)\le 1$.
Nov 12, 2012 at 10:20 comment added Nima Thank you for the answer. I will think about what extra condition we need in the first case. Please let me know if you know any reasonable condition.
Nov 12, 2012 at 10:17 vote accept Nima
Nov 12, 2012 at 10:11 history answered Qing Liu CC BY-SA 3.0