Is exceptional divisor always a Projective bundle over the centre?

Question

Let $f:\tilde X \rightarrow X$ be a blow up at center Z. Is $f^{-1}(z)=\mathbb{P}^{k}$ ?for some $k$, $\forall$ $z\in Z$

In the case of blow-up of $\mathbb{A}^{n}$ at origin it is very clear that the $f^{-1}(0)=\mathbb{P}^{n-1}$.

This is the description of blowing up an affine variety centered at a closed sub-variety.

In this case also the fibers over the center are $\mathbb{P}^{k}$.

My Question,

1.is it true that in a general blow-up of a scheme centered at closed sub-scheme the fibers over the center is a projective space $\mathbb{P}^{k}$

2. If Yes, How do we find what is the Projective space?

3. If yes, Is the exceptional divisor over the center is a projective bundle. (i.e., is it locally trivial).