# Blow up of 9 points in 3-fold and intersection of strict transforms

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.