I want to prove that a function $\pi : \mathbb{C}_{*}^{n}\mapsto \mathbb{C}^{n}$ is bijective. Where $\mathbb{C}_{*}^{n}$ is the explosion of $\mathbb{C}^{n}$ and is defined as
$\mathbb{C}_{*}^{n}:= \{(z,l) \in \mathbb{C}^{n}\times\mathbb{C}P^{n-1} | z \in l\}$
and $l$ is a line that pases through the origin in $\mathbb{C}^{n}$
thank you, I am not very acquainted with the explosion over a field, can you explain me.
Presumably, $\pi(z,\ell)=z$? It's not bijective, at least for $n>1$.
$$\pi^{-1}(0)=\{(0,\ell)\mid\ell\in\mathbb CP^{n-1}\}$$
It is a bijection when excluding the $0\in \mathbb C^n$ and $\pi^{-1}(0)$ from the sets, because when $z\in\mathbb C^n\setminus\{0\}$, there is exactly one line $\ell$ through zero containing $z$.