There are already several questions on the subject, the most interesting being this one. But I do not find the answers there completely satisfying. The end of the proof is missing.
Here is how I see the main steps of the proof. Let $X$ be a compact connected Riemann surface of genus $g$.
Prove the Riemann-Roch theorem for $X$. I am aware that this part is quite hard. It needs some serious work in analysis/fuctional analysis to establish even the existence of one non-constant meromorphic function, and still further non-trivial work to prove the Riemann-Roch theorem. But this is well-documented, for instance in Foster's book "Lectures on Riemann SUrfaces" or Gunning's book "Riemann Surfaces" or Miranda's "algebraic curves and Riemann Surfaces".
Use Riemann-Roch to show that a line bundle of degree $\leq 2g+1$ defines an embedding of the curve into $\mathbb{P}^N$ for some $N$. This is easier, and the proof for Riemann surfaces is exactly the same as for algebraic curves, so to the references above we can add Harthsorne II.7 and IV.3, etc.
Conclude. Now this is the part where I ask for some help and reference. It is not done in any of the reference I have cited (not even in Miranda's, despite its title).
Here is how I can see part 3. Using any divisor of degree at least $2g+1$ (e.g. $2g+1$ times a point), we get an embedding of the curve $X$ into $\mathbb{P}^N$. Call $Y$ its image. Show that $Y$ is an analytic subvariety of $\mathbb{P}^N$. Then use Chow's theorem, using that $Y$ is compact since $X$ is, to prove that $Y$ is an algebraic curve in $\mathbb{P}^N$.
However, this is a long and hard road. Chow's theorem is a difficult and long to prove theorem, either with its original proof or with Serre's GAGA (since GAGA is itself quite advanced). It requires introducing the notion of analytic subvarieties, which I'd like to avoid. I am looking for the simplest possible proof of this, as this is for a graduate course I am teaching. Is there some shorter way to do this? Is there a reference that does it?
