= Nowhere-vanishing section of a globally generated bundle on a curve
Let $F$ be globally generated of rank $r\ge2$ 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.
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