# 代數曲線

## 射影曲線

$g_i(x_1, \ldots, x_n) \longrightarrow (X_0)^{\deg g_i} g_i\left(\frac{X_1}{X_0}, \ldots, \frac{X_n}{X_0}\right)$

## 代數函數域

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

## 奇點

### 判斷方式

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

### 奇點分類

x3 = y2

$f(x,y) = \sum_{n \geq 1} f_n(x,y)$

$f_m(x,y) = \prod _{i=1}^m (a_i x - b_i y)$

$g = \frac{1}{2}(d-1)(d-2) - \sum_P \delta_P,$

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

$\mu = 2\delta -r + 1$

## 曲線的例子

### 有理曲線

$F$上的有理曲線雙有理等價於射影直線$\mathbb{P}^1_F$的曲線，換言之，其函數域同構於單變元有理函數域$F(t)$。當$F$代數封閉時，這也等價於該曲線之虧格為零，對一般的域則不然；實數域上由$x^2+y^2+1=0$給出的函數域虧格為零，而非有理函數域。

x2 + xy + y2 = 1

。考慮斜橢圓$E: x^2+xy+y^2=1$，其中$(-1,0)$是有理點。畫一條過該點且斜率為t之直線$y=t(x+1)$，並帶入E的等式，於是得到：

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

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

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

### 橢圓曲線

$y^2z + a_1 xyz + a_3 yz^2 = x^3 + a_2 x^2z + a_4 xz^2 + a_6 z^3. \,\!$

## 文獻

• 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