The Stacks project

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 on

This lemma needs to be moved earlier because the result is used silently in Lemmas 45.14.3, 45.14.4, and 45.14.15. I supposed we could move it and Lemma 37.17.1 to their own section in some section later than Section 37.16. Then in the proof we cannot use the computations from Section 50.16, so these would have to be replicated in a local remark or something (don't change Section 50.16).


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