# Meromorphic section of a given line bundle over a compact Riemann surface

Let $$\Sigma$$ be a compact Riemann surface and $$L \to \Sigma$$ be a given(!) line bundle, with $$c_1(L) \geq N(P) > 0$$ where $$N(P)$$, depending on $$P$$ as soon to be explained, is an integer as large as you desire, but also fixed.
I want to know wether $$L$$ admits a meromorphic section with exactly $$P$$ poles, counting multiplicity. If you can prescribe them, it would be even better, but it is no must.

I am aware of the fact that you can come up with a meromorphic section having at least $$1$$ pole (not necesarrily simple) at a prescribed point (see Forster, lectures on Riemann surfaces theorem 29.16). That being said, we can pick $$P$$ distinct points, say $$a_1,...,a_P \in \Sigma$$, which would then give sections $$s_i: \Sigma \to L$$ having a pole precisely at $$a_i$$ (order unnknown, might be double or triple) and define $$s=\sum_{i=1}^P s_i$$
which gives me a meromorphic section with at least $$P$$ poles. In this case, the information on the chern class would just yield the number of zeroes $$N$$ via the formula $$N-P=\langle c_1(L), \Sigma \rangle$$, a completely worthless information for my purpose. I am aware of many closeley related theorems (e.g. Riemann Roch, which only gives me meromorphic functions/Holomorphic section) and the "pole problem" is solveable in the complex plane. I feel like this should be a well-studied problem, since thanks to my chern class/degree of my line bundle being high enough (say higher than genus of $$\Sigma$$ plus some correction term) I should be able to deduce it from a known theorem.

I am grateful for any reference/hint anyone could give me.

First, recall that Riemann-Roch states that $$l(D)-l(K-D)=\deg D + 1 -g$$. Combining this with the fact that a divisor of negative degree has no global sections, we observe that once $$\deg D > \deg K$$ we get $$l(K-D)=0$$, and so $$l(D+P)=l(D)+1$$ for any point $$P$$.
We can use this to solve the problem as follows. For a fixed divisor $$D$$ with $$\deg D > \deg K+1$$, we let $$S=\{D'\in\operatorname{Div} \Sigma \mid D' \text{ effective and } \exists P\in\Sigma \text{ with } D'+P=D\}.$$ Then for each $$D'\in S$$, we see that $$l(D')=l(D)-1$$, and as $$S$$ is finite, the union of the vector spaces $$\mathcal{L}(D')$$ inside $$\mathcal{L}(D)$$ cannot be the whole space. Therefore there are meromorphic functions in $$\mathcal{L}(D)$$ which have poles exactly on $$D$$.
• @F.Conrad I think the key idea to keep in mind for this kind of problem is that meromorphic sections of $\mathcal L$ with poles along a divisor $D$ correspond to holomorphic sections of the twist $\mathcal L(D)$. Nov 24, 2022 at 1:43