TITLE: Line subbundles of vector bundles on smooth curves
QUESTION [5 upvotes]: Let $V$ be a vector bundle of rank at least 2 on a smooth (integral, projective) curve $C$. We know that a global section of $V$ is the same as a morphism $\mathcal{O}_C \to \mathcal{V}$ (letting $\mathcal{V}$ denote the sheaf of sections), and that this section is nonzero if and only if the morphism of sheaves is injective.
On the other hand, let $\mathbb{P}(V)$ be the associated projective bundle. A nonzero section $s : C \to V$ defines a rational map to $\mathbb{P}(V)$. Since $C$ is a smooth curve and $\mathbb{P}(V)$ is proper, this extends to an actual morphism $C \to \mathbb{P}(V)$, and so yields an actual line-subbundle $L \subset V$, and (again) an injective morphism of sheaves $\mathcal{L} \to \mathcal{V}$.
Question: How can we obtain the line bundle $L$ (and its sheaf of sections) in sheaf language? e.g. is there an algebraic way of converting the morphism $\mathcal{O}_C \to \mathcal{V}$ into the morphism $\mathcal{L} \to \mathcal{V}$ ? Also, what is $c_1(L)$ in this case? (edit: I guess $c_1(L) = Z(s)$, the section's vanishing locus?)

REPLY [2 votes]: Ok, I have a somewhat ad-hoc construction. The key algebraic fact about smooth curves (to use instead of the fact that rational morphisms from smooth curves to proper varieties extend to genuine morphisms) is that a coherent sheaf on $C$ is locally free if and only if it is torsionfree. This is because $C$ is covered by affine charts which are the spectra of Dedekind domains.
Let $\varphi_s : \mathcal{O}_C \to \mathcal{V}$ be the section and let $\mathcal{F} = \mathrm{coker}(\varphi_s)$. Let $\mathcal{F}_{tors}$ be the torsion subsheaf, which is supported on $Z(s)$. So
$$0 \to \mathcal{F}_{tors} \to \mathcal{F} \to \mathcal{F}' \to 0$$
and $\mathcal{F}'$ is torsionfree, hence locally free. Let $\mathcal{L} \subset \mathcal{V}$ be the preimage of $\mathcal{F}_{tors}$ (note that $\mathrm{im}(\varphi_s)$ is a subsheaf of $\mathcal{L}$). We have
$$0 \to \mathcal{L} \to \mathcal{V} \to \mathcal{F}' \to 0,$$
which shows that $\mathcal{L}$ is locally free, and
$$0 \to \mathcal{O}_C \to \mathcal{L} \to \mathcal{F}_{tors} \to 0,$$
which shows that $\mathcal{L}$ has rank 1 and that $c_1(L) = Z(s)$.