I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why?
We know that the Birkhoff-Grothendieck Theorem says that any bundle on $\mathbb{P}^1$ must be a direct sum of line bundles. So any subbundle of $E$ must look like $\oplus_i^r \mathcal{O}(a_i) $ (where $r\leq k$). So what are the constraints on the $a_i$? Naively, I'd like to say that the $a_i$ just have to be 1 to match the decomposition of $E$, but I don't think this is true.
And the same question, but more general would be what are the subbundles of $\oplus_i^k \mathcal{O}(b_i) $. This would probably be more useful to more people.