The Stacks project

Definition 54.5.1. Let $X$ be a scheme such that every quasi-compact open has finitely many irreducible components. Let $x \in X$ be a closed point. The normalized blowup of $X$ at $x$ is the composition $X'' \to X' \to X$ where $X' \to X$ is the blowup of $X$ in $x$ and $X'' \to X'$ is the normalization of $X'$.