Let $X$ be a smooth projective curve, $\mathcal{F}$ a torsion sheaf on $X$, $\Gamma(X,\mathcal{F})$ finitely generated by $ \{s_1,\ldots,s_n\}$ (or $n:= \dim \Gamma(X, \mathcal{F}) $). Is $\mathcal{F}$ globally generated? That is, is the map $\bigoplus_{i=1}^n\mathcal{O}_{X,p}\to \mathcal{F}_p$ surjective for every $p\in X$?
This is clearly the case when $p$ is generic, as $\mathcal{F}$ is torsion ($\mathcal{F}_p = 0$). And there are only finitely many the support is zero-dimensional, but how do I know what happens in these cases?