Riemann Surfaces of Algebraic Functions

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

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Motivation

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

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

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

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Smooth irreducible affine plane curve

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

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

then $g$ 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 $f\in\mathbb{C}[z,,w]$, $f\neq 0$. Define $Z(f):=f^{-1}(0)$ to be an affine plane curve. Write

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

Define the degree of $f$ to be

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

and the multiplicity of $f$ at $0$ to be $

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

and for $p=(a,b)\in\mathbb{C}\times\mathbb{C}$, define the multiplicity of $f$ at $p$ to be

$$\mathrm{mult}_{p}(f):=\mathrm{mult}_0(f(z-a,w-b))$$

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

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

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

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

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

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

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

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

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

$$T_pX=Z\left((z-a)\frac{\partial f}{\partial z}(p) + (w-b)\frac{\partial f}{\partial w}(p) \right).$$

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 $X$ is a Riemann surface.

Proof. See other articles.$\square$

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Nonsingular projective plane curves

The main object of this section is projective space.

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

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

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

$$\mathbb{P}^1\hookrightarrow\mathbb{P}^2\hookrightarrow\cdots$$

where

$$\mathbb{P}^1=\mathbb{C}^1\cup\infty$$

$$\mathbb{P}^2=\mathbb{C}^2\cup\mathbb{P}^1=\mathbb{C}^2\cup\mathbb{C}^1\cup\infty.$$

More generally,

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

Def. The homogenization of a function $f\in\mathbb{C}[x_1,\ldots,x_n]$ of degree $d$ is

$$f^*:=x_0^df(\frac{x_1}{x_0},\frac{x_2}{x_0},\ldots,\frac{x_n}{x_0})\in\mathbb{C}[x_0,\ldots,x_n]_d.$$

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

$$F_*:=F(1,x_1,\ldots,x_n)\in\mathbb{C}[x_1,\ldots,x_n].$$

The arithmetic properties of homogenization and dehomogenization are clear.

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

$$F(z,w)=\prod_{i=1}^k(a_iz+b_iw)^{r_i}$$

for some $a_i,b_i\in\mathbb{C}$, $r_i\in\mathbb{Z}$, and $\sum r_i=d$​​.

Def. Let $F\in\mathbb{C}[x_0,\ldots,x_n]_d$ be a homogeneous polynomial of degree $d$. If the only solution to

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

is

$$(x_0,\ldots,x_n)=(0,\ldots,0),$$

then $F$​ is said to be nonsingular.

Now the case are down to Riemann surface (dimension $1$), and the number of variables $z_0,\ldots,z_n$ is $3$.

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

Def. Let $F\in\mathbb{C}[z_0,z_1,z_2]_d$ be a nonconstant homogeneous polynomial. Define $X=Z(F)$ to be a projective plane curve. Moreover, if $F$ is nonsingular, then $X$ 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 $F\in\mathbb{C}[z_0,z_1,z_2]_d$ be a nonsingular homogeneous polynomial of degree $d$. Then $X=Z(F)\subset\mathbb{P}^2$​ is a Riemann surface.

Proof. See other articles. $\square$​

An additional result about projective plane curve is as follows.

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

$$\bar{X}:=Z(f^*)\subset\mathbb{P}^2$$

be the projective closure of $X$​.

Theorem. $|\bar{X}-X|<\infty$.

Proof. Decompose

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

Then the homogenization of $f$ can be written as

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

Thus

$$\{F=0,z_0=0\}=\{f_d=0,z_0=0\}$$

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

Since $f_d\in\mathbb{C}[z_1,z_2]_d$, then $f_d=\prod_{i=1}^k(z_1-\alpha_iz_2)^{r_i}$, then

$$|\bar{X}\cap\{z_0=0\}|=|Z(f_d)|=k<\infty$$

Therefore,

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

This completes the proof. $\square$

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