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?