# Properties of the contraction map of the exceptional divisor of a blow-up

Let $$X$$ be a smooth variety and $$Z$$ be a smooth subvariety. Consider the blow-up at $$Z$$ $$\pi:\mathrm{Bl}_Z(X)\rightarrow X$$ and let $$E$$ be the exceptional divisor. What are the properties of the restriction map $$\pi|_E:E\rightarrow Z$$?

Of course it is projective, more explicitly, if $$Z$$ is given by the coherent ideal $$\mathcal{I}$$ of $$X$$ then the map is given by $$E=\underline{\mathrm{Proj}}_Z(\oplus_{n\geq 0}\mathcal{I}^n/\mathcal{I}^{n+1})\rightarrow Z.$$ That is, it is the projective morphism given by the associated graded ring of the ideal $$\mathcal{I}$$.

Nevertheless, I think that this map should be much more than just projective in general. For example, I would like to know if this map is flat or smooth or if it has any other property relating it to a projection of the form $$W\times Z\rightarrow Z$$.

If more generally you know something about the general contraction of the exceptional locus of a birrational proper/projective map between smooth varieties it would be really nice! Thanks in advance.

• $E$ is a projective bundle over $Z$; it is the projectivisation of either the normal or conormal bundle of $Z$ in $X$, depending on your choice of convention. In particular it is smooth. – Pop May 22 at 15:07
• Sorry, I meant to write that $\pi_{|E}: E \rightarrow Z$ is smooth. – Pop May 22 at 15:39