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.


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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.