I have the surface
$W=Z(x_0x_1-x_2x_3)$, in $\mathbb{P^3}$
and I want to describe it as a union of an affine piece and some other piece laying in $\mathbb{P^2}$. My solution is to look at:
1-$W\cap D(x_0)$
2-$W\cap Z(x_0)$
regrading (1) I have its the surface laying in $\mathbb{A^3}$ giving by zero loci of $f=x-yz$ regrading (2) the intersection contains three points $(0:1:0:0),(0:0:1:0),(0:0:0:1),(0:1:0:1),(0:1:1:0)$ now can I conclude that the second piece is isomorphic to $\mathbb{P^2}$?
$W \cap Z(x_0)$ is the intersection of two surfaces in $\mathbb{P}^3$, so it should be a curve. In particular, we're looking at points $[0:x_1:x_2:x_3]$ such that $x_2 x_3 = 0$. Of course this implies either $x_2 = 0$ or $x_3 = 0$, so we're looking at the union of the two lines $[0:\ast:0:\ast]$ and $[0:\ast:\ast:0]$.