For positive integers $a_0,\dots,a_n$, consider the weighted projective space $\Bbb C\Bbb P(a_0,a_1,\dots,a_n)$, which is the quotient of $\Bbb C^{n+1}-\{0\}$ by the action of $\Bbb C^*=\Bbb C-\{0\}$ defined by $t\cdot (z_0,\dots,z_n)=(t^{a_0}z_0,\dots,t^{a_n}z_n)$.

According to https://en.wikipedia.org/wiki/Weighted_projective_space, the weighted projective space $\Bbb C\Bbb P(a_0,a_1,\dots,a_n)$ is isomorphic to the quotient of projective space $\Bbb {CP}^n$ by the group that is the product of the groups of roots of unity of orders $a_0,a_1,\dots,a_n$ acting diagonally. I am trying to verify this in the simple case $n=2$. For $m\geq1$ let $\mu_m$ be the group of roots of unity of order $m$, and for $a,b,c\geq 1$ let $G=\mu_a\times \mu_b\times \mu_c$. There is a well-defined diagonal action of $G$ on $\Bbb {CP}^2$. I have to show that $\Bbb {CP}^2/G=\Bbb {CP}(a,b,c)$. Maybe one way to do this is to define a map $\Bbb {CP}^2\to \Bbb {CP}(a,b,c)$ and descend to the quotient. But the map $\Bbb {CP}^2\to \Bbb {CP}(a,b,c)$, $[x,y,z]\to [x,y,z]$ is not well-defined and I got stuck. Any hints?


1 Answer 1


Suppose $\rho_a$ is a primtive $a$-th root of unity, and likeiwse for $\rho_b$ and $\rho_c$. Define an action of $\mu_a\times \mu_b\times \mu_c$ on $\mathbb{C}P^2$ as follows. For an element $(x,y,z)\in \mu_a\times \mu_b\times \mu_c$ and an element $[z_0:z_1:z_2]\in \mathbb{C}P^2$, we set $$(x,y,z)\ast [z_0:z_1:z_2] = [\rho_a^x z_0: \rho_b^y z_1: \rho_c^z z_2].$$

Define a map $\mathbb{C}P^2\rightarrow X:=\mathbb{C}P(a,b,c)$ by $f([z_0:z_1:z_2])\rightarrow [z_0^a: z_1^b: z_2^c]$.

I claim this is well defined. Indeed, for $\lambda \in \mathbb{C}\setminus\{0\}$, we have $$f([\lambda z_1: \lambda z_2:\lambda z_3]) = [\lambda^a z_0^a: \lambda^b z_1^b: \lambda^c z_2^c] = [z_0^a:z_1^b:z_2^c] = f([z_0:z_1:z_2]).$$

Moreover, it's not too hard to see that $f$ is invariant under the action of $\mu_a\times \mu_b\times \mu_c$, so we get an induced map $\mathbb{C}P^2/(\mu_a\times \mu_b\times \mu_c)\rightarrow \mathbb{C}P(a,b,c)$. Can you prove that this map is a homeomorphism?


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