I want to know whether it is possible to bound the degree of line subbundles of certain holomorphic line bundles over Riemann surfaces.
Even more concretely, consider the complex projective line $\mathbb{P}_{\mathbb{C}}^1$ and the rank-2 vector bundle $E=\mathcal{O}(a)\oplus \mathcal{O}(b)$ for $a,b\in\mathbb{Z}$. Let $L\subset E$ be a holomorphic line subbundle. Thanks to the classification of line bundles over the Riemann sphere, this is some $\mathcal{O}(d)$ for $d\in\mathbb{Z}$.
Question: is necessarily $d$ bounded by $a,b$ or $a+b=deg(E)$? How can I prove that? Is there a maximal-degree subbundle of $E$?