Riemann Surfaces of Algebraic Functions

In this part we shall consider the Riemann surfaces of algebraic functions.


The prototype of vanishing algebraic functions should be the graph of a holomorphic function. Let DCD\subset\mathbb{C} be a domain, gO(D)g\in\mathcal{O}(D). Define

X:={(z,g(z))C×C:zD}C×CX:=\{(z,g(z))\in\mathbb{C}\times\mathbb{C}:z\in D\}\subset \mathbb{C}\times\mathbb{C}

endowed with subspace topology. There is a natural projection π:XD\pi:X\rightarrow D, which is also a homeomorphism with continuous inverse π1(z)=(z,g(z))\pi^{-1}(z)=(z,g(z)). Thus π\pi is a global complex chart on XX, making XX a Riemann surface.

Smooth irreducible affine plane curve

Def. Let DCD\subset\mathbb{C} be a domain, gO(D)g\in\mathcal{O}(D). If fC[z,w]\exists f\in\mathbb{C}[z,w], f0f\neq 0 ,such that

f(z,g(z))=0,zD,f(z,g(z))=0,\quad\forall z\in D,

then gg is said to be algebraic.

We shall discuss the space of zeros of algebraic functions, which are Riemann surfaces. The simplest space of algebraic functions are affine plane curves.

Def. Let fC[z,,w]f\in\mathbb{C}[z,,w], f0f\neq 0. Define Z(f):=f1(0)Z(f):=f^{-1}(0) to be an affine plane curve. Write

f=i,jaijziwj.f=\sum_{i,j}a_{ij}z^i w^j.

Define the degree of ff to be

deg(f):=max{i+j:aij0}\deg(f):=\max\{i+j:a_{ij}\neq 0\}

and the multiplicity of ff at 00 to be $

mult0(f):=min{i+jaij0}\mathrm{mult}_0(f):=\min\{i+j|a_{ij}\neq 0\}

and for p=(a,b)C×Cp=(a,b)\in\mathbb{C}\times\mathbb{C}, define the multiplicity of ff at pp to be


With the notation of multiplicity, we can determine whether a zero of ff is regular or singular.

Def. Let fC[z,,w]f\in\mathbb{C}[z,,w], f0f\neq 0 and pZ(f)p\in Z(f).

  1. If multp(f)=1\mathrm{mult}_p(f)=1, then pp is said to be regular.

  2. If multp(f)>1\mathrm{mult}_p(f)>1, then pp is said to be singular.

Moreover, denote S={pX:multp(f)>1}S=\{p\in X:\mathrm{mult}_p(f)>1\} to be the singular set of ff. In particular, in the case that ff is a polynomial with two variables, SS can be written as

S=Z(f,fz,fw)S=Z(f,\frac{\partial f}{\partial z},\frac{\partial f}{\partial w})

Def. If S=S=\emptyset, then ff is said to be smooth. (All zeros are regular)

Def. If ff cannot be written as the product of two polynomials, then ff is reducible.

Cor. If ff is irreducible in C[z,w]\mathbb{C}[z,w], then

TpX=Z((za)fz(p)+(wb)fw(p)).T_pX=Z\left((z-a)\frac{\partial f}{\partial z}(p) + (w-b)\frac{\partial f}{\partial w}(p) \right).

The following is the main result of this section, showing that IT IS WORTHY RESEARCHING RIEMANN SURFACES!!! However, the proof is a little bit long, and it requires another post to present.

Theorem. A smooth irreducible affine plane curve XX is a Riemann surface.

Proof. See other articles.\square

Nonsingular projective plane curves

The main object of this section is projective space.

Fact. The nn-dimensional complex projective plane Pn\mathbb{P}^n is a compact complex manifold of dimension nn.

Remark. There exists a biholomorphism CˉP1\bar{\mathbb{C}}\cong\mathbb{P}^1.

Remark. The projective plane can be regarded as a complex of inclusion





More generally,

Pn=knCk.\mathbb{P}^n=\bigsqcup_{k\leq n}\mathbb{C}^k.

Def. The homogenization of a function fC[x1,,xn]f\in\mathbb{C}[x_1,\ldots,x_n] of degree dd is


Similarly, the dehomogenization of a homogeneous function FC[x0,,xn]dF\in\mathbb{C}[x_0,\ldots,x_n]_d is defined to be


The arithmetic properties of homogenization and dehomogenization are clear.

Cor. Up to some power of x0x_0, factoring FC[x0,,xn]dF\in\mathbb{C}[x_0,\ldots,x_n]_d is the same as factoring FC[x1,,xn]F_*\in\mathbb{C}[x_1,\ldots,x_n]. In particular, if FC[z,w]dF\in\mathbb{C}[z,w]_d, then


for some ai,biCa_i,b_i\in\mathbb{C}, riZr_i\in\mathbb{Z}, and ri=d\sum r_i=d​​.

Def. Let FC[x0,,xn]dF\in\mathbb{C}[x_0,\ldots,x_n]_d be a homogeneous polynomial of degree dd. If the only solution to

{F=0Fxi=0,i=0,1,,n\begin{cases} F&=0\\ \frac{\partial F}{\partial x_i}&=0,\quad i=0,1,\ldots,n \end{cases}



then FF​ is said to be nonsingular.

Now the case are down to Riemann surface (dimension 11), and the number of variables z0,,znz_0,\ldots,z_n is 33.

Remark. As it is stated is Algebraic Geometry, FC[x,y,z]dF\in\mathbb{C}[x,y,z]_d is not a well-defined function on P2\mathbb{P}^2, while the zeros Z(F)Z(F) is well-defined on P2\mathbb{P}^2​.

Def. Let FC[z0,z1,z2]dF\in\mathbb{C}[z_0,z_1,z_2]_d be a nonconstant homogeneous polynomial. Define X=Z(F)X=Z(F) to be a projective plane curve. Moreover, if FF is nonsingular, then XX is said to be smooth.

Similar to the case of AFFINE PLANE CURVE, the following theorem shows that smooth projective plane curves are Riemann surface.

Theorem. Let FC[z0,z1,z2]dF\in\mathbb{C}[z_0,z_1,z_2]_d be a nonsingular homogeneous polynomial of degree dd. Then X=Z(F)P2X=Z(F)\subset\mathbb{P}^2​ is a Riemann surface.

Proof. See other articles. \square

An additional result about projective plane curve is as follows.

Def. Let fC[z1,z2]f\in\mathbb{C}[z_1,z_2] and X=Z(f)C2X=Z(f)\subset\mathbb{C}^2 be an affine plane curve. Define


be the projective closure of XX​.

Theorem. XˉX<|\bar{X}-X|<\infty.

Proof. Decompose

f=i=0dfi,deg(fi)=i.f=\sum_{i=0}^d f_i, \quad\deg(f_i)=i.

Then the homogenization of ff can be written as

F:=f=i=0dx0difi.F:=f^*=\sum_{i=0}^d x_0^{d-i}f_i.



Xˉ{z0=0}={[0,z1,z2]P2fd(z1,z2)=0}\Rightarrow \bar{X}\cap\{z_0=0\}=\left\{[0,z_1,z_2]\in\mathbb{P}^2|f_d(z_1,z_2)=0\right\}

Since fdC[z1,z2]df_d\in\mathbb{C}[z_1,z_2]_d, then fd=i=1k(z1αiz2)rif_d=\prod_{i=1}^k(z_1-\alpha_iz_2)^{r_i}, then



Xˉ={F=0,z00}{F=0,z0=0}=Xfinitey many points\begin{aligned} \bar{X}&=\{F=0,z_0\neq 0\}\cup\{F=0,z_0=0\}\\ &=X\cup\text{finitey many points} \end{aligned}

This completes the proof. \square