Definition 31.34.1. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. A morphism $X' \to X$ is called a $U$-admissible blowup if there exists a closed immersion $Z \to X$ of finite presentation with $Z$ disjoint from $U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$.
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