Definition 31.32.1 (01OG)—The Stacks project

Definition 31.32.1. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subscheme corresponding to $\mathcal{I}$, see Schemes, Definition 26.10.2. The blowing up of $X$ along $Z$, or the blowing up of $X$ in the ideal sheaf $\mathcal{I}$ is the morphism

\[ b : \underline{\text{Proj}}_ X \left(\bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n\right) \longrightarrow X \]

The exceptional divisor of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the center of the blowup.

Comments (1)

Comment #10413 by Sándor Kovács on July 10, 2025 at 21:53

This definition of the exceptional divisor is only correct if $Z$ is everywhere at least codimension $2$ . A more precise definition is that the exceptional divisor is the preimage of $Z\setminus Z_{\rm Cartier}$ where $Z_{\rm Cartier}\subseteq Z$ is the largest open subset of $Z$ which is locally a Cartier divisor. This actually matters when one blows up a Weil divisor to make it Cartier.

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7 comment(s) on Section 31.32: Blowing up

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