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The Stacks project

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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