Birational map between manifolds

Question

I have to show that the manifold $A=\{ [z_{1}:z_{2}:z_{3}:z_{4}]\in \mathbb{C}\mathbb{P}^{3} | z_{1}z_{3}^{n} - z_{2}z_{4}^{n}=0\}$ is birational equivalent to $\mathbb{C}\mathbb{P}^{2}$, how can I found a birational map between them?

Answer 1

First a small remark: $A$ isn't a manifold if $n>1$. It is singular along the line $z_3=z_4=0$, as you can check by the Jacobian criterion.

To answer the question: try the rational map $A \dashrightarrow \mathbf P^2$ defined by $[z_1,z_2,z_3,z_4] \mapsto [z_2,z_3,z_4]$.