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.

There are also:

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