Suppose we have blown up a variety $X$ at some points $P_j$ so that we introduce exceptional divisors $E_j$ in $\widetilde X$; what is the general strategy to determine the intersections of these exceptional divisors with, say, the strict transform of a divisor on $X$ containing some of the $P_j$?

Here's a more specific context.

Let $X$ be a 3-fold (for the sake of simplicity say over $\mathbb{C}$) and assume that there is a divisor $X_0 \subseteq X$ such that $X_0 = Z_1 \cup \ldots \cup Z_4$.

Assume that the only singular points of $X$ are 9 ordinary nodes $P_1,\ldots ,P_4$ such that each Z_j contains four nodes and $P_i,P_j\in Z_i\cap Z_j$ for $i\neq j$.

Now I blow up these nine points and I get a 3-fold $\widetilde X$ with exceptional divisors $E_j\simeq \mathbb{P}^2$.

1) How can I calculate the strict transforms $\widetilde Z_j$ of $Z_j$ if I know, for example, that $Z_j\simeq \mathbb{P}^1\times \mathbb{P}^1$?

2) What can I say about the intersections $\widetilde Z_j \cap \widetilde Z_k$ and $\widetilde E_i \cap \widetilde Z_j$?

I know that $\widetilde Z_j$ is the projectivisation of the normal bundle of $Z_j$ in $X$ but am not really confident working with this kind of arguments.

A good reference (even multiple books) to gain hands-on experience on determining the geometry of blow ups would also be really appreciated.


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