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Say $F$ is a globally generated vector bundle on $X$ of rank $f$. Let $s$ be a nowhere vanishing global section of $F$. Why do we obtain a short exact sequence $$0 \to \mathcal{O}_X \to F \to V \to 0,$$ where $V$ is spanned vector bundle of rank $f-1$?

How can one prove this? Thanks in advance.

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Note that $\mathcal{O}_X$ is (isomorphic to) the trivial line subbundle of $F$ spanned by $s$, and $V$ is the quotient of $F$ by $\mathcal{O}_X$, i.e. $V = F/\mathcal{O}_X$, so $\dim V = \dim (F/\mathcal{O}_X) = \dim F - \dim \mathcal{O}_X = \dim F - 1$.

We did not need to assume that $F$ was globally generated.

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  • $\begingroup$ Thank you, Mr. Albanese. I conjectured this already, but I wasn't sure. $\endgroup$ – L_K666 Aug 18 '16 at 20:12

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