求出曲线的奇点

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$F\left(x_0, x_1, x_2\right)$是一个齐次多项式.
在$ℝℙ^2$中的曲线$$𝒞=\{\left[a_0, a_1, a_2\right]\mid F\left(a_0, a_1, a_2\right)=0\}$$在点$𝐚=\left[a_0, a_1, a_2\right]$处,若$\nabla F(𝐚)≠𝟎$,切线为$$x_0 \frac{∂F}{∂x_0}(𝐚)+x_1 \frac{∂F}{∂x_1}(𝐚)+x_2 \frac{∂F}{∂x_2}(𝐚)=0$$若$\nabla F(𝐚)=𝟎$, 则$𝐚$为曲线$𝒞$的奇点.

求出$y^2=x^3$和$y^2=x^2(x+1)$和$y=x^3$的奇点

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$𝐚=[x,y,z]$
$F(𝐚)=y^2z-x^3=0$
$\nabla F(𝐚)=[-3x^2,2yz,y^2]=𝟎\implies 𝐚=[0,0,1]$ is the point $(0,0)$ on $\mathbb R^2$.

从图中看，在 0 处，切线是多重的。
$F(𝐚)=y^2z-x^2(x+z)=0$
$\nabla F(𝐚)=[-3x^2-2xz,2yz,y^2-x^2]=𝟎\implies 𝐚=[0,0,1]$ is the point $(0,0)$ on $\mathbb R^2$.

从图中看，在 0 处，切线不是唯一的。
$F(𝐚)=x^3-yz^2=0$
$\nabla F(𝐚)=[-x (3 x + 2 z), 2 y z, -x^2 + y^2]=𝟎\implies 𝐚=[0,1,0]$ is on the line at infinity.

从图中不能看出它在无穷远处有一个奇点（而 $y=x^2$ 没有奇点）。

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Example 3.2. Let $K$ be any field of characteristic two and let $\mathcal{X}$ be the curve defined over $K$ by $F=Y^2 Z-X^3+X^2 Z$. Then $F_X=X^2, F_Y=0, F_Z=Y^2+X^2=(Y-X)^2$. Hence $P=(a: b: c)$ is singular if and only if $a=0, b=a$, that is $P=(0: 0: 1)$ is the only singular point of $\mathcal{X}$.