= Upper degree bound for line subbundles on a smooth curve
Choose a rational basis of the dual of a <vector bundle> and an <effective divisor> $D$ bounding all its poles. Evaluation gives a generically <injective> map $E\to\mathcal O_C(D)^{\oplus r}$. A <line subbundle> maps nontrivially to at least one summand, so $\mathcal O_C(D)\otimes L^{-1}$ has a nonzero regular section. Consequently $\deg L\le\deg D$. This bound depends on $E$ and the chosen rational frame, and does not require $E$ to be stable.
Back to article page
