The Stacks project

Definition 89.8.2. Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral algebraic space over $S$. We say $Y$ has a resolution of singularities by normalized blowups if there exists a sequence

\[ Y_ n \to X_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y \]

where

  1. $Y_ i$ is proper over $Y$ for $i = 0, \ldots , n$,

  2. $Y_0 \to Y$ is the normalization,

  3. $Y_ i \to Y_{i - 1}$ is a normalized blowup for $i = 1, \ldots , n$, and

  4. $Y_ n$ is regular.

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View Definition 89.8.2 as pdf