Let's take $\phi:\mathcal{X}\rightarrow B$ a family of deformations of complex surfaces above the one dimensional disk $B=D\_r(0)\subset \mathbb{C}$. Suppose that $X:=\phi^{-1}(0)$ has one ordinary double point $p$, while all other fibers are smooth. So we can take $x,y,z$ complex coordinates around $p$ such that $\phi(x,y,z)=x^2+y^2+z^2$.


Now take the function $m:\widetilde{B}:=D\_{\frac{1}{2}}(0)\rightarrow B$, $m(t)=t^2$ and consider the deformation $\widetilde{\mathcal{X}}=\mathcal{X}\times\_B \widetilde{B}$. If $\widetilde{p}\in \widetilde{\mathcal{X}}$ is the only point mapping to $p$ then in local coordinates the germ of $\widetilde{\mathcal{X}}$ in $\widetilde{p}$ is isomorphic to the germ in the ordigin of $V(x^2+y^2+z^2-t^2)\subset \mathbb{C}^4$.


I consider the blow up of $\widetilde{\mathcal{X}}$ in $\widetilde{p}$: its exceptional divisor $E$ is a quadric in $\mathbb{P}^3$ and so is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ by Segre embedding. Finally it shoul come out that $N\_{E/\widetilde{\mathcal{X}}}\simeq\mathcal{O}\_{\mathbb{P}^1}(-1)\boxtimes\mathcal{O}\_{\mathbb{P}^1}(-1)$ and $\mathcal{O}\_{\mathbb{P}^1}(-1)\boxtimes\mathcal{O}\_{\mathbb{P}^1}(-1)=p\_1^\*\mathcal{O}\_{\mathbb{P}^1}(-1)\otimes p\_2^\*\mathcal{O}\_{\mathbb{P}^1}(-1)$ where $p\_1$ and $p\_2$ are the two projections from $\mathbb{P}^1\times\mathbb{P}^1$)


My questions are:


1) I have read that for example for the blow up of the origin in $\mathbb{A}^2$, the normal bundle of the exceptional divisor $\mathbb{P}^1$ is $\mathcal{O}\_{\mathbb{P}^1}(-1)$, but why is that?


2) In the specific case of my construction in which way can i prove that $N\_{E/\widetilde{\mathcal{X}}}$ is actually the box product?



Normal bundle of an exceptional divisor
For the first question, and what Anton mentions, that the normal bundle of the exceptional divisor of the blow up of a smooth subvariety **in a smooth variety** is $\mathscr O(-1)$ see Thm 8.24 in [Hartshorne]. This takes care of your first question. 


However, you have to be careful, because 


1. this fails if the ambient scheme is singular, as in your case, where $\widetilde{\mathcal{X}}$ is singular, and
2. you have to keep in mind that the $(-1)$ here means a specific embedding, so it might not translate to "the natural" $(-1)$ of any given scheme, say $\mathbb P^n$ (this actually does not happen in your case, but only by luck).


To see that the normal bundle you are looking for (I assume, because what you wrote makes no sense) is the box product do this:


First let's give names to our players: Let $A$ denote the blow up of $\mathbb A^4$ at the origin, $P\simeq \mathbb P^3$ the exceptional divisor, $B\subset A$ the blow up of $\widetilde{\mathcal{X}}$ at the same point, and $E$ the exceptional divisor of that. Presumably you want $N\_{E/B}$.


Observe that $A$ is smooth and hence all divisors are Cartier. Then consider the restrictions of $\mathscr O\_A(P)$ to both $P$ and $B$ and then the restrictions of these to $E$:


$$
\mathscr O\_A(P)|\_P \simeq N\_{P/A}\simeq \mathscr O\_{\mathbb P^3}(-1)
$$


$$
\left(\mathscr O\_A(P)|\_B\right)|\_E \simeq \mathscr O\_B(E)|\_E \simeq N\_{E/B}
$$


and hence


$$
N\_{E/B}\simeq \mathscr O\_A(P)|\_E \simeq \mathscr O\_{\mathbb P^3}(-1)|\_E\simeq \mathscr O\_{\mathbb P^1}(-1)\boxtimes \mathscr O\_{\mathbb P^1}(-1).
$$


