# 代数曲线

## 射影曲线

${\displaystyle g_{i}(x_{1},\ldots ,x_{n})\longrightarrow (X_{0})^{\deg g_{i}}g_{i}\left({\frac {X_{1}}{X_{0}}},\ldots ,{\frac {X_{n}}{X_{0}}}\right)}$

## 代数函数域

${\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x^{2}+y^{2}+1=0\}=\emptyset }$

[(P=c) end if=0]

[(P=c) end if=0]

## 奇点

N-1n+1=0

### 判断方式

${\displaystyle {\frac {\partial f}{\partial x}}(P)={\frac {\partial f}{\partial y}}(P)={\frac {\partial f}{\partial z}}(P)=0\quad (P\in C)}$

### 奇点分类

x3 = y2

${\displaystyle f(x,y)=\sum _{n\geq 1}f_{n}(x,y)}$

${\displaystyle f_{m}(x,y)=\prod _{i=1}^{m}(a_{i}x-b_{i}y)}$

${\displaystyle g={\frac {1}{2}}(d-1)(d-2)-\sum _{P}\delta _{P},}$

${\displaystyle \nabla f(x,y):B_{\epsilon }-\{(0,0)\}\rightarrow B_{\epsilon }-\{(0,0)\}}$

${\displaystyle \mu =2\delta -r+1}$

## 曲线的例子

### 有理曲线

${\displaystyle F}$上的有理曲线双有理等价于射影直线${\displaystyle \mathbb {P} _{F}^{1}}$的曲线，换言之，其函数域同构于单变元有理函数域${\displaystyle F(t)}$。当${\displaystyle F}$代数封闭时，这也等价于该曲线之亏格为零，对一般的域则不然；实数域上由${\displaystyle x^{2}+y^{2}+1=0}$给出的函数域亏格为零，而非有理函数域。

x2 + xy + y2 = 1

。考虑斜椭圆${\displaystyle E:x^{2}+xy+y^{2}=1}$，其中${\displaystyle (-1,0)}$是有理点。画一条过该点且斜率为t之直线${\displaystyle y=t(x+1)}$，并带入E的等式，于是得到：

${\displaystyle x={\frac {1-t^{2}}{1+t+t^{2}}}}$
${\displaystyle y=t(x+1)={\frac {t(t+2)}{1+t+t^{2}}}}$

${\displaystyle X^{2}+XY+Y^{2}=Z^{2}\,\!}$

${\displaystyle X=1-t^{2},\quad Y=t(t+2),\quad Z=t^{2}+t+1\,\!}$

### 椭圆曲线

${\displaystyle y^{2}z+a_{1}xyz+a_{3}yz^{2}=x^{3}+a_{2}x^{2}z+a_{4}xz^{2}+a_{6}z^{3}.\,\!}$

## 文献

• Egbert Brieskorn and Horst Knörrer, Plane Algebraic Curves, John Stillwell, trans., Birkhäuser, 1986
• Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, American Mathematical Society, Mathematical Surveys Number VI, 1951
• Hershel M. Farkas and Irwin Kra, Riemann Surfaces, Springer, 1980
• Phillip A. Griffiths, Introduction to Algebraic Curves, Kuniko Weltin, trans., American Mathematical Society, Translation of Mathematical Monographs volume 70, 1985 revision
• Robin Hartshorne, Algebraic Geometry, Springer, 1977
• Shigeru Iitaka, Algebraic Geometry: An Introduction to the Birational Geometry of Algebraic Varieties, Springer, 1982
• John Milnor, Singular Points of Complex Hypersurfaces, Princeton University Press, 1968
• George Salmon, Higher Plane Curves, Third Edition, G. E. Stechert & Co., 1934
• Jean-Pierre Serre, Algebraic Groups and Class Fields, Springer, 1988