Definition 71.17.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subspace corresponding to $\mathcal{I}$ (Morphisms of Spaces, Lemma 67.13.1). 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.
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