I am trying to solve the following problem:
Let $X$ be a compact Riemann surface, and $p\in X$ a point. Let $\Omega(2p)$ be the complex vector space of meromorphic $1$-forms on $X$ that have (possibly) a pole of order at most $2$ at $p$, and are holomorphic elsewhere. Show that $$\dim \Omega(2p)=g+1.$$
(Do not use the Riemann-Roch Theorem)
My attempt:
Since we need to avoid Riemann-Roch theorem, I want to write the basis of the vector space explicitly.
All l know is that due to residue theorem on a compact Riemann surface, we have the sum of residue is $0$. So we have two kinds of meromorphic 1-form:
- holomorphic everywhere
- $p$ is the pole of order two, and near it, the meromorphic 1-form can be written as $$\frac{c_{-2}}{z^2}dz+\sum_{n=0}^{\infty}c_n z^ndz$$
Then I don't know how to continue. Could anyone give any hint or comments?
I am also curious what role the genus plays in this problem.
The only thing we need to prove is that there is exactly one meromorphic 1-form on the Riemann surfaces with a pole at $p$ of order 2 and locally looks like $$\frac{1}{z^2}+\text{analytic part}.$$
For the above statement, there is a proof in this notes: Chap II, Cor 3.3.