非奇异非空二次曲面有多少个射影等价类

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Proposition 44证明了$\mathbb RP^2$中非奇异非空二次曲线共有1个射影等价类.
首先合同对角化得到$a_0x_0^2+a_1x_1^2+a_2x_2^2=0$, 其中$a_0,a_1,a_2∈\{0,1\}$

全正和全负对应空集. 剩下2种.
因为同乘$-1$不改变方程,共有$2/2=1$个等价类.

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$\mathbb RP^3$中非奇异非空二次曲面共有2个射影等价类. 首先合同对角化得到$a_0x_0^2+a_1x_1^2+a_2x_2^2+a_3x_3^2=0$, 其中$a_0,a_1,a_2,a_3∈\{0,1\}$

全正和全负对应空集. 剩下3种.
因为同乘$-1$不改变方程,共有$\lceil3/2\rceil=2$个等价类:$+++-$和$++--$
类似可证明$\mathbb RP^n$中非奇异非空quadric有$\left\lceil n\over2\right\rceil$个等价类?

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$+++-$ 椭球、双叶双曲面
$++--$ 单叶双曲面、双曲抛物面
刚才想到,单叶双曲面、双曲抛物面是doubly ruled surface,而椭球、双叶双曲面不是.

The properties of being ruled or doubly ruled are preserved by projective maps, and therefore are concepts of projective geometry.

看来,$\mathbb RP^3$中非奇异非空二次曲面有2类是合理的

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Topological classification of hyper-quadrics

Fundamental results in affine and projective geometry imply that every real or complex hyper-quadric in n-space is affinely or projectively equivalent to an example from a finite and reasonably short list of examples.

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2. Show that the number of projective equivalence classes of hypequadrics in $\mathbb{R} \mathbb{P}^n$ is equal to $\frac{1}{4}(n+2)(n+4)$ if $n$ is even and $\frac{1}{4}(n+3)^2$ if $n$ is odd.

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Math 631: Problem Set 4

a). Prove that any two non-singular quadrics in $\mathbf P^n$ are projectively equivalent