TITLE: Embedding Riemann surfaces into $\mathbb P^2$
QUESTION [6 upvotes]: Suppose I am given a Riemann surface $\Sigma_g$ of genus $g$. What is known about the sufficient and necessary conditions needed on $\Sigma_g$ to have an embedding into $\mathbb P^2$?
If $\mathcal T_g$ is the Teichmuller space of genus $g$ Riemann surfaces, is there any way to intrinsically describe the subspace of plane curves?

REPLY [5 votes]: If I am not mistaken, for the exceptional values of the genus, the locus of degree $d$ plane curves has (complex) dimension
$$
\frac{(d+2)(d+1)}{2}-9
$$
sitting inside a moduli space of dimension
$$
\frac{3(d-1)(d-2)}{2}-3
$$
so it becomes relatively small as $d$ increases.
This article by Landesman seems to be pertinent for your question.