Let $a_0, \dots, a_n$ be positive integers with $\gcd = 1$. Consider the weighted projective space $\mathbb {CP}^n[a_0, \dots, a_n]$. I have used the following procedure to construct a coordinate chart centered at an arbitrary point $[b_0 : \dots : b_k] \in \mathbb{CP}^n[a_0, \dots, a_n]$.
After permuting the coordinates, we may assume that $b_i \ne 0$ for every $i = 0, \dots, k$, whereas $b_i = 0$ for every $i = k+1, \dots, n$. We will construct a coordinate chart for the open subset of $\mathbb {CP}^n[a_0, \dots, a_n]$ whose preimage is $(\mathbb C^\star)^{k+1} \times \mathbb C^{n-k}$.
Equip $U = (\mathbb C^\star)^{k+1}$ with the $\mathbb C^\star$ action by weighted rescaling
$$\lambda \cdot (z_0, \dots, z_k) = (\lambda^{a_0} z_0, \dots, \lambda^{a_k} z_k)$$
Equip $V = \mathbb C^{n-k}$ with the $\mathbb C^\star$ action by weighted rescaling
$$\mu \cdot (z_{k+1}, \dots, z_n) = (\mu^{a_{k+1}} z_{k+1}, \dots, \mu^{a_n} z_n)$$
Let $G \subset \mathbb C^\star$ be the subgroup of $a$-th roots of $1$, where $a = \gcd(a_0, \dots, a_k)$. Notice that $G$ acts trivially on $U$, hence the “true acting group” is $\mathbb C^\star / G$. On the other hand, $G$ acts effectively on $V$, because $\gcd(a, a_{k+1}, \dots, a_n) = 1$, so our next step is to trivialize this action.
Let $W = \mathbb C^{n-k} / G$, equipped with the trivial action of $\mathbb C^\star / G$. Define the map $\varphi : U \times V \to U \times W$ by
$$\varphi(z_0, \dots, z_n) = (z_0, \dots, z_k, \mu^{a_{k+1}} z_{k+1}, \dots, \mu^{a_n} z_n)$$
where the auxiliary variable $\mu \in \mathbb C^\star / G$ satisfies $\mu^a z_0 = 1$. A direct calculation shows that
$$\lambda \circ \varphi(z_0, \dots, z_n) = (\lambda^{a_0} z_0, \dots, \lambda^{a_k} z_k, \mu^{a_{k+1}} z_{k+1}, \dots, \mu^{a_n} z_n)$$
and a slightly less direct calculation shows that
$$\varphi \circ \lambda(z_0, \dots, z_n) = (\lambda^{a_0} z_0, \dots, \lambda^{a_k} z_k, \mu^{a_{k+1}} z_{k+1}, \dots, \mu^{a_n} z_n)$$
as well. Hence $\varphi$ is $\mathbb C^\star$-equivariant.
Now consider the map induced by $\varphi$ between the orbit spaces
$$\tilde \varphi : \frac {U \times V} {\mathbb C^\star} \longrightarrow \frac {U \times W} {\mathbb C^\star}$$
Since $\mathbb C^\star$ acts locally freely on $U$ and trivially on $W$, we have an isomorphism
$$\frac {U \times W} {\mathbb C^\star} \cong (\mathbb C^\star)^k \times W$$
We shall show that $\tilde \varphi$ is an isomorphism. For starters, $\tilde \varphi$ is surjective by construction, hence $\tilde \varphi$ is surjective as well. However, $\varphi$ is not necessarily injective. To work around this issue, notice that every orbit in $U \times V$ passes through $a_0$ points of the form $(1, z_1, \dots, z_n)$, which we will call the orbit's canonical representatives. Now write explicitly
$$\varphi(1, z_1, \dots, z_n) = (1, z_1, \dots, z_k, \eta z_{k+1}, \dots, \eta z_n)$$
where $\eta \in \mathbb C^\star / G$ is the identity element. Since $a$ divides $a_0$, every $a$-th root of $1$ is also an $a_0$-th root of $1$. Thus, any two canonical representatives that are set to the same orbit in $U \times W$ differ by at most a weighted rotation by an $a_0$-th root of $1$. Hence they represent the same orbit in $U \times V$. Hence $\tilde \varphi$ is injective.
My questions are:
Does my procedure for constructing charts centered at any point of $\mathbb {CP}^n[a_0, \dots, a_n]$ actually work?
How do I show that these charts are compatible? Actually, what does it even mean for two charts on an orbifold to be compatible?
Assuming that the charts are compatible, I think I have shown that $[b_0 : \dots : b_n]$ is a regular point of $\mathbb {CP}^n[a_0, \dots, a_n]$ if and only if
$$1 = \gcd \{ a_i \text{'s such that } b_i \ne 0 \}$$
Otherwise, no neighborhood of this point is biholomorphic to a polydisc $\mathbb D^n$. However, I vaguely recall reading somewhere (although I cannot find the link right now) that an orbifold's underlying topological space could be homeomorphic to a topological manifold. That is, we could have orbifold singularities in the smooth structure, even though the underlying topological space is locally homeomorphic to $\mathbb R^n$. How do I prove that this does not happen in $\mathbb {CP}^n[a_0, \dots, a_n]$? That is, how do I prove that $\mathbb {CP}^n[a_0, \dots, a_n]$ is not even locally homeomorphic to $\mathbb C^n$ at the points with orbifold singularities?