Lemma 37.17.3. Let $S$ be a scheme. Let $Z \to X$ be a closed immersion of schemes smooth over $S$. Let $b : X' \to X$ be the blowing up of $Z$ with exceptional divisor $E \subset X'$. Then $X'$ and $E$ are smooth over $S$. The morphism $p : E \to Z$ is canonically isomorphic to the projective space bundle
\[ \mathbf{P}(\mathcal{I}/\mathcal{I}^2) \longrightarrow Z \]
where $\mathcal{I} \subset \mathcal{O}_ X$ is the ideal sheaf of $Z$. The relative $\mathcal{O}_ E(1)$ coming from the projective space bundle structure is isomorphic to the restriction of $\mathcal{O}_{X'}(-E)$ to $E$.
Proof.
By Divisors, Lemma 31.22.11 the immersion $Z \to X$ is a regular immmersion, hence the ideal sheaf $\mathcal{I}$ is of finite type, hence $b$ is a projective morphism with relatively ample invertible sheaf $\mathcal{O}_{X'}(1) = \mathcal{O}_{X'}(-E)$, see Divisors, Lemmas 31.32.4 and 31.32.13. The canonical map $\mathcal{I} \to b_*\mathcal{O}_{X'}(1)$ gives a closed immersion
\[ X' \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ X}(\mathcal{I})\right) \]
by the very construction of the blowup. The restriction of this morphism to $E$ gives a canonical map
\[ E \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ Z}(\mathcal{I}/\mathcal{I}^2)\right) \]
over $Z$. Since $\mathcal{I}/\mathcal{I}^2$ is finite locally free if this canonical map is an isomorphism, then the final part of the lemma holds. Having said all of this, now the question is étale local on $X$. Namely, blowing up commutes with flat base change by Divisors, Lemma 31.32.3 and we can check smoothness after precomposing with a surjective étale morphism. Thus by the étale local structure of a closed immersion of schemes over $S$ given in Lemma 37.17.1 this reduces us to the case discussed in Remark 37.17.2.
$\square$
Comments (2)
Comment #7503 by Johan on
Comment #7643 by Stacks Project on