 Nowhere-vanishing section of a globally generated bundle on a curve

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.