One can explicitly describe the above compactification as follows. For any point $p\in\Sigma$ there is a neighborhood $U\subset Y$ of $p$ such that $U\cong\mathbb{R}^{3}\times\mathbb{C}^{n-2}$ and $U\cap\Sigma$ can be identified with $\{0\}\times\mathbb{C}^{n-2}$. By unicity of $\pi$, there is a commutative diagram

$$\begin{CD}\pi^{-1}(U)@>{\cong}>{}>\mathbb{C}^{2}\times\mathbb{C}^{n-2}\\ @V{\pi}V{}V@V{\bar{\pi}}V{}V\\ U@>{\cong}>{}>\mathbb{R}^{3}\times\mathbb{C}^{n-2}\end{CD}$$

(1)

where $\bar{\pi}(z_{1},z_{2},\zeta)=(|z_{1}|^{2}-|z_{2}|^{2},z_{1}z_{2},\zeta)$, $\zeta\in\mathbb{C}^{n-2}$.