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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.

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5 Answers 5

up vote 6 down vote accepted

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.

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The reason for P being so special in genus 1 - in my opinion at least - is that {1,P,P'} generate the ring of meromorphic functions on the curve (see e.g. Alain Robert's book). This goes against Riemann-Roch in higher genera. –  David Lehavi Oct 27 '09 at 23:24
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@David Speyer: this is great, I was looking for an answer along these lines. Do you know of a good reference that goes into this in more detail? Also do you know if you restrict genus g = 1/2(d-1)(d-2) for d a positive integer if its possible to find an embedding and not just an immersion into P^2? –  solbap Oct 28 '09 at 1:05
    
@administrator: The moduli of degree d curves is the number of monomials of degree d in 3 variables minus the dimension of GL_3, all in all: (d+2)(d+1)/2 - 9 (don't expect this to work for d < 3: too many automorphisms). Which is much smaller then 3g-3. –  David Lehavi Oct 28 '09 at 10:08

For $X = \Delta/\Gamma$ a compact Riemann surface of genus $g>1$ a good analogue of the Weierstrass $p$-function is the Poincaré series $f_a(z) = \sum_{\gamma \in \Gamma} \gamma^*(q_a)$, where $q_a = dz^2/(z-a)$ is a quadratic differential with a simple pole at $a \in \Delta$. This series gives a meromorphic quadratic differential on $X$ with a simple pole at any desired point and no other singularities. If $b$ is close enough to $a$ then $f_b/f_a$ gives a meromorphic function on $X$ with a simple zero at $a$, providing local coordinates. By compactness one can then choose finitely many $a_1,\ldots,a_n$ such that the corresponding $f_1,\ldots,f_n$ give an embedding of $X$ into $P^{n-1}$.

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Professor, welcome to the site! –  Bombyx mori May 20 '13 at 6:36

The genus g of a curve in P^2 is

g = 1/2(d-1)(d-2)

where d is the degree of the homogeneous polynomial that cuts it out. So you can't have a Riemann surface of genus, say, 4, sitting inside P^2.

(This is in Hartshorne Chapter I.7, although you have to be willing to believe that complex Riemann surfaces "are" non-singular projective curves over C and "arithmetic genus" is "number of holes.")

If you move to P^3, such gadgets do exist (you have holomorphic maps Universal Cover ---> Riemann Surface ---> P^3). But they're not going to be meromorphic functions anymore, since P^2 is special in that "meromorphic function" = "map to P^2."

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Perhaps my post led to a misunderstanding. I did not specify the dimension of the projective space; for a surface of high genus the embedding is initially into a projective space of high dimension. You will need enough meromorphic functions to realize it as an embedded nonsingular subvariety. This you can get by taking various rational functions of two such meromorphic functions analogous to pe and pe'. Afterwards getting it into a projective space of low dimension is, as others have pointed out, an issue of algebraic geometry. Also, the meromorphic functions I talk about are meromorphic functions on the Riemann surface; i. e. holomorphic mappings from the surface into the Riemann sphere a.k.a. the extended complex plane a.k.a. one-dimensional complex projective space.

But Shafarevich explains this much better than I.

The question of explicit embedding obviously depends on how the Riemann surface is actually given to you concretely. If it is given by a Fuchsian group acting on the unit disk, you can embed it using Poincare series. These are analogues of the series used to define the Weierstrass pe-function.

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In general the answer to 2) will be no since not every Riemann surface of genus >1 embeds into P^2 - the best one can do in general is find a curve birational to it with nodal singularities obtained by projecting from an embedding in P^3.

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