plane curve and compact Riemann surface

Riemann surfaces are one dimension complex manifold. Yet, apart from the elementary examples, like \mathbb{P}^1,\mathbb{C},\mathbb{H} and their quotients by some groups, it is not easy to construct directly some compact Riemann surfaces(of course, according to the Uniformization theorem, all compact Riemann surfaces arise in as quotients of these three Riemann surfaces). Here we get inspirations from the theory of manifolds. In fact, we know that some important examples of manifolds arise as particular subsets of the Euclidean spaces. In particular, we can construct some manifolds from the loci of some polynomials. This is the way we are going to use in the following.

For a simple of dimension, we consider a polynomial of two variables non constant, P\in \mathbb{C}[X,Y]. We write it as P(X,Y)=a_0(X)Y^d+a_1(X)Y^{d-1}+...+a_d(X) where a_0(X) is not the zero polynomial. We set C_P=\{(x,y)\in\mathbb{C}^2|P(x,y)=0\}, the zero set of P. For the present, we pretend that C_P is a Riemann surface(we can see that, it is not a Riemann surface because of some points). Then we should try to find these ‘bad’ points. One way is to find a holomorphic map from C_P to some Riemann surface that we know, and use this map to find which points are not ‘good’.

Consider the projection \pi_1:C_P\rightarrow \mathbb{C},(x,y)\mapsto x. Note that, for those x\in\mathbb{C} such that a_0(x)\neq 0, we have that P_x(Y)=P(x,Y) has d solutions for the variable Y. So, if we note S=\{x\in\mathbb{C}|a_0(x)=0\}, so \pi_1: C_P-\pi_1^{-1}(S)\rightarrow \mathbb{C}-S is a ramified covering of degree d. so, in this way C_P'=C_P-\pi_1^{-1}(S) is a Riemann surface.

If a_0(X) is a non-zero constant, then S=\emptyset, we can say something more. In fact, we can compactify \mathbb{C} to \mathbb{P}^1, and add also one point to C_P to make it compact. So, in this way, we get a compact Riemann surface. In this case, we can even calculate the genus of C_P using Riemann-Hurwitz formula.

Here we consider a simple example. Suppose that P(X,Y)=Y^d-\prod_{i=1}^l(X-x_i) where x_i are distincts points and gcd(d,l)=1. Then, the ramified points of \pi_1 are x_1,...,x_l,\infty of ramification degree d(the point difficult is the infinity). So, according to the Riemann-Hurwitz formula, we have that

2-2g(C_P\bigcup \{\infty\})=d(2-2g(\mathbb{P}^1))-(d-1)(l+1)

Thus we get that

g(C_P\bigcup \{\infty\})=(l-1)(d-1)/2.

This formula shows that, for any genus g, there is a polynomial such that the Riemann surface defined by this polynomial(with compactification) is of genus g.

This method is, in some sense, more explicit than the quotient method.

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