= Unbounded negative degrees of line subbundles
{title2=$\mathcal O_C(-mp_0)\subset E\quad(m\gg0)$}
For a rank-two bundle $E$, the sufficiently positive twist $E(mp_0)$ is globally generated. The <nowhere-vanishing section of a globally generated bundle on a curve> gives a subbundle $\mathcal O_C\subset E(mp_0)$. Twisting back produces a genuine <line subbundle> of degree $-m$, with <locally free> quotient. Thus line-subbundle degrees are bounded above but not below. Merely inserting a negative <divisor on an algebraic curve> into a fixed subline would allow a torsion quotient and would not establish this stronger conclusion.
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