# Possible subbundles on $\mathbb{CP}^1$

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.

• I think you should explain what you mean by subbundle. Do you mean $\mathscr F\subseteq E$ such that $\mathscr F$ is locally free, or do you also require that $\mathscr E/\mathscr F$ is locally free? I personally think that the former should be called locally free subsheaf and only the latter deserves to be called subbundle. The reason for this is that in topology, any [injective] map $\mathcal O_{\mathbb P^1}(-1) \to \mathcal O_{\mathbb P^1}$ does not realise $\mathcal O(-1)$ as a subbundle of $\mathcal O$ (since the subspace you get does not have the same dimensions at all points).
– Remy
Jan 20, 2016 at 23:36
• @Remy Fair point. I had never thought of that. I mean your first possibility: $\mathscr{F}\subset E$ such that $\mathscr{F}$ is locally free. Jan 21, 2016 at 15:20

While the general question can be answered, the answer is not so nice, so let me concentrate on the first case of $E=\mathcal{O}(1)^k$. As you said, $r\leq k$ and if $r=k$, $E$ is the only subbundle. So, assume that $r<k$. Then of course, $a_i\leq 1$ and this is the only constraint. That is, any $\oplus_{i=1}^r \mathcal{O}(a_i)$ with $r<k, a_i\leq 1$ can be realized as a subbundle of $E$. The proof is essentially an application of Serre's theorem (and induction on $r$) which says that a globally generated vector bundle of rank greater than the dimension of the variety has a nowhere vanishing section. You can see a proof in Mumford's book on Curves on a surface, for example.
• Thanks for the help. I'd like to look at the proof. I just got the book from the library and am trying to find the right section. Is it "Lecture 7: Resume of the cohomology of coherent sheaves on $\mathbb{P}^n$"? Jan 20, 2016 at 19:00