There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
Engelbrekt gave an overview of how you start with a compact Riemann surfaces and map them into projective space Links between Riemann surfaces and algebraic geometry
In the case of a genus 1 surface X there's a very explicit construction. Namely X can be realized as ℂ/L for a lattice L ≅ ℤxℤ. From here the Weierstrass p function and its derivative can be constructed
and these give you a map ℂ/L --> ℙ^2 via z |--> [p(z), p'(z), 1] which realizes X as a degree three curve in ℙ^2
Say now X is a compact Riemann surface of genus g > 1. As has been pointed out below I should restrict to say g = 1/2(d-1)(d-2) where d>3, because otherwise there is no hope to realize X as a nonsingular curve in ℙ^2.
1) a complex manifold Y that is a covering space of X such that X ≅ Y/G where G is the covering group of Y over X
2) holomorphic functions f₁, f₂, f₃ from Y/G to ℂ∪∞
such that z |--> [f₁,(z), f₂(z), f₃(z)] realizes X as a projective variety of dimension 1 in ℙ^2?
I'm told a good choice for Y would be the hyperbolic plane because then the 4g-gon representation of a genus g surface tiles the plane.