= Global generation of positive twists on a smooth curve
{title2=$E\otimes\mathcal O_C(mp_0)\text{ is globally generated for }m\gg0$}
For a <vector bundle> of finite positive rank let $B$ be an upper degree bound for the <line subbundles> of $E^*\otimes K_C$. If $m-1>B$, no nonzero map $\mathcal O_C(mp_0-p)\to E^*\otimes K_C$ can exist for any point $p$: its saturated image would have degree at least $m-1$. <Serre duality> then gives $H^1(E(mp_0-p))=0$. The evaluation <exact sequence> shows that $H^0(E(mp_0))\to E(mp_0)|_p$ is <surjective> at every point, proving the assertion with a uniform bound in $p$.
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