Definition 89.4.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a closed point. By Decent Spaces, Lemma 68.14.6 we can represent $x$ by a closed immersion $i : \mathop{\mathrm{Spec}}(k) \to X$. The blowing up $X' \to X$ of $X$ at $x$ means the blowing up of $X$ in the closed subspace $Z = i(\mathop{\mathrm{Spec}}(k)) \subset X$.
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