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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.

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    $\begingroup$ $E$ 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. $\endgroup$ – Pop May 22 at 15:07
  • $\begingroup$ Sorry, I meant to write that $\pi_{|E}: E \rightarrow Z$ is smooth. $\endgroup$ – Pop May 22 at 15:39

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