I am trying to understand something that for me is familiar fact in the study of 4-manifolds, however I am trying to understand it through the lens of algebraic geometry (which I sadly know very very little about). The familiar fact is that $(S^2 x S^2) \sharp \overline{\mathbb{C}P^2}$ is diffeomorphic to $\mathbb{C}P^2 \sharp \overline{\mathbb{C}P^2} \sharp \overline{\mathbb{C}P^2}$. I've always thought of this as a fun first exercise in Kirby calculus and a complex analog of the result that $(S^1 \times S^1) \sharp \mathbb{R} P^2 = 3 \mathbb{R}P^2$.
I was reading this article by Milnor where he proves that the prime decomposition of a 3-manifold is unique and a the end of the article he mentions the above diffeomorphism to illustrate how this uniqueness does not generalize to higher dimensions. He also mentions a proof of this isomorphism due to Hirzebruch and it is this proof that I was hoping to get some help with.
First, we had better put our algebraic geometer hats on, so $S^2 \times S^2 = \mathbb{C}P^1 \times \mathbb{C}P^1$. Now we are supposed to look at the expression $$ (z_0 : z_1, z_2) \leftrightarrow (z_0 : z_1), (z_0 : z_2) $$ I imagine we are supposed to think that this is a pair of functions \begin{align*} \mathbb{C}P^2 - \{(0:1:0), (0:0:1) \} &\to \mathbb{C}P^1 \times \mathbb{C}P^1 \\ (z_0 : z_1 : z_2) &\mapsto ((z_0:z_1), (z_0:z_2)) \end{align*} and \begin{align*} \mathbb{C}P^1 \times \mathbb{C}P^1 - {((0:1),(0:1))} &\to \mathbb{C}P^2\\ ((z_0:z_1), (w_0:w_1)) &\mapsto (z_0 : z_1 : w_1 \frac{z_0}{w_0}) \text{ if } w_0 \neq 0\\ &\mapsto (w_0 : z_1 \frac{w_0}{z_0} : w_1) \text{ if } z_0 \neq 0\\ \end{align*} and note that these are inverses of each other, at least where composition makes sense.
Ok, so now what am I supposed to do? Milnor says the word "blow-up" (which unfortunately doesn't mean anything to me) and I guess each of those missing points is going to contribute to a $\overline{\mathbb{C}P^2}$ summand being added. Can someone explain this to me?
I imagine the same expression but where everything above is done over the real numbers gives a proof of the familiar fact about surfaces mentioned above.