I am sorry for the sloppyness of the question. I assumed, without stating, that the matrix had linear entries. Now I've edited it.

This is probably known, but just to draw a line at the end of this post: I think that, at least generically, the normal bundle is $\mathcal{O}\oplus \mathcal{O}$ and it is sent onto the quadric surface $Q$ in $P^3$ in which $X$ is a divisor of type $(2,3)$.

Thank you, this is the construction I was looking for. Can the normal sheaf of the line $L_o \subset X$ be both $\mathcal{O}\oplus \mathcal{O}$ and $\mathcal{O}(1)\oplus \mathcal{O}(-1)$?

Yes, you're of course right. Anyway my question may still stand in the following terms. There could be a birational model of $X$ (say a blow-up) that dominates both $X$ and one of the varieties I mentioned in my question.

@ Roy: In fact it is, as you pointed out, a question of minimal model theory. The word "relatively" (that was a bit confusing to me) just means that your objects live above a positive dimensional subscheme of the base.

Hi Charles, thank you for your answers and references, and congrats on your blog: I am a fan! Luckily enough, I managed to answer my own questions regarding the two ancient questions, whereas this one is still quite open. It is a pleasure discussing about maths and, since it is likely that we will go a bit off topic, I'll write you an email asap.