There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
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
http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions
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
QUESTION:
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.
Is there
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.
Best Answer
I think Hunter and Greg's answers make it hard to see the forest for the trees. Let X be a compact Riem. surface of genus >= g. Let Y be the universal cover of X equipped with the complex structure pulled back from X. As a complex manifold, Y is isomorphic to the upper half plane, and the deck transformations form a subgroup Gamma of PSL_2(R). There will be characters chi of Gamma for which there are nonzero functions f on Y such that f(gz) = chi(g) f(z). For chi ample enough (not defined here), we will be able to choose functions (f_1, f_2, f_3) such that z --> (f_1(z) : f_2(z) : f_3(z)) gives an immersion X --> P^2. All of this works in any genus.
The technical issue is that this map is an immersion, not an injection, meaning that the image can pass through itself. One can either decide to live with this, or work with maps to P^3 instead.
Most books that I have seen don't lift all the way to the universal cover of X. Instead, they take the covering of X which corresponds to the commutator subgroup of pi_1(X). This can be motivated in a particular nice way in terms of the Jacobian. This is a complex manifold with the topological structure of a 2g dimensional torus. There is a map X --> J, so that the map pi_1(X) --> pi_1(J) is precisely the map from pi_1(X) to its abelianization. People then work with the universal cover of J, and the preimage of X inside it. This has three advantages: the universal cover of J is C^g, not the upper half plane; the group of Deck transformations is Z^{2g}, not the fundamental group of a surface, and the action on C^g is by traslations, not Mobius transformations. The functions which transform by characters, in this setting, are called Theta functions*, and they are given by explicit Fourier series.
*This is a slight lie. Theta functions come from a certain central extension of the group of Deck transformations. It is certain ratios of Theta functions that will transform by characters as sketched above. The P function itself, for example, is a ratio of four Theta functions. In the higher genus case, in my limited reading, I haven't seen names for these ratios, only for the Theta functions.