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Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ an auxiliary closed point. Can we find for all $m\geq 0$, a section $f\in H^{0}(X,E\otimes O_{X}(D_{m}))$

with $D_{m}=[x_{1}]+\dots+[x_{n}]+m[z]$ such that at the points $x_{1},\dots,x_{n}$ the section $f$ factorizes through the subbundle $E'$?

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Dear prochet, I think the question may need some extra conditions to admit a reasonable answer. As it stands, the answer seems to be either trivially yes, if the zero section is an acceptable answer, or trivially no, if the bundle E tensor O(D_m) doesn't have any nonzero global sections, e.g. if m < -deg E - n. –  Artie Prendergast-Smith Feb 13 '13 at 19:07
    
edit to last comment: I meant to write, if E is a line bundle with m < -deg E -n. –  Artie Prendergast-Smith Feb 13 '13 at 19:08
    
Yes I shouldn't have formulate like this. In fact, I only need the proposition for $V'\subset V$ two vector spaces such that $E$ (resp E') are the trivial vector bundles of fibres V and V' –  prochet Feb 13 '13 at 19:18
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