Globally generated vector bundle ($H^0(C,F)\otimes {\mathcal{O}}_C\twoheadrightarrow F$)

A vector bundle is globally generated when its global sections span every fibre. The evaluation map in the display expresses this condition. On a projective curve its space of global sections is finite-dimensional, so a finite collection of sections suffices.

For a vector bundle of finite positive rank let $B$ be an upper degree bound for the line subbundles of $E^{\ast }\otimes K_C$. If $m-1>B$, no nonzero map ${\mathcal{O}}_C(m{p}_0-p)\to E^{\ast }\otimes K_C$ can exist for any point $p$: its saturated image would have degree at least $m-1$. Serre duality then gives $H^1(E(m{p}_0-p))=0$. The evaluation exact sequence shows that $H^0(E(m{p}_0))\to E(m{p}_0){|}_p$ is surjective at every point, proving the assertion with a uniform bound in $p$.

Let $F$ be globally generated of rank $r\ge 2$ on a projective curve, and put $W=H^0(C,F)$. The pairs $(p,s)$ with $s(p)=0$ form the total space of the kernel of the evaluation map over the curve. This incidence variety has dimension $\dim W+1-r<\dim W$. Its image in $W$ is closed because the curve is proper, and is a proper subset by the dimension bound. A section outside it vanishes nowhere and gives a line subbundle ${\mathcal{O}}_C\subset F$ with locally free quotient.