Suppose $X$ is a Noetherian, separated, nonsingular integral scheme of dimension $1$.
In this post, I learned that if $V$ is a vector bundle (locally free sheaf) ove $X$, then any nonzero global section $s$ gives a line subbundle $L$ (means the line bundle $L\subset V$, and $V/L$ is still a locally free sheaf). But what if $V$ has no nonzero global sections? Can we still get a line subbundle of $V$?
Any help is appreciated thanks!
Best Answer
A week later, let's put this to rest.
The key observation, as Tabes Bridges notes in the comments, is that schemes of dimension one have nice properties. In fact, any noetherian separated scheme of dimension one has an ample invertible sheaf $\mathcal{L}$, see Stacks 09N7. In particular, this means that for any quasi-coherent sheaf of finite type $\mathcal{F}$ on $X$ there is an integer $n_0$ so that for all $n\geq n_0$, $\mathcal{F}\otimes\mathcal{L}^{\otimes n}$ is globally generated. Therefore we may solve the problem for $V\otimes \mathcal{L}^{\otimes n}$, obtain a sub line bundle, and then twist back to end up with a sub line bundle of $V$.