The Stacks project

87.5 Dominating by normalized blowups

In this section we prove that a modification of a surface can be dominated by a sequence of normalized blowups in points.

Definition 87.5.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$ satisfying the equivalent conditions of Morphisms of Spaces, Lemma 65.49.1. 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 $X$ at $x$ (Definition 87.4.1) and $X'' \to X'$ is the normalization of $X'$.

Here the normalization $X'' \to X'$ is defined as the algebraic space $X'$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 65.49.1 by Divisors on Spaces, Lemma 69.17.8. See Morphisms of Spaces, Definition 65.49.3 for the definition of the normalization.

In general the normalized blowing up need not be proper even when $X$ is Noetherian. Recall that an algebraic space is Nagata if it has an ├ętale covering by affines which are spectra of Nagata rings (Properties of Spaces, Definition 64.7.2 and Remark 64.7.3 and Properties, Definition 28.13.1).

Lemma 87.5.2. In Definition 87.5.1 if $X$ is Nagata, then the normalized blowing up of $X$ at $x$ is a normal Nagata algebraic space proper over $X$.

Proof. The blowup morphism $X' \to X$ is proper (as $X$ is locally Noetherian we may apply Divisors on Spaces, Lemma 69.17.11). Thus $X'$ is Nagata (Morphisms of Spaces, Lemma 65.26.1). Therefore the normalization $X'' \to X'$ is finite (Morphisms of Spaces, Lemma 65.49.6) and we conclude that $X'' \to X$ is proper as well (Morphisms of Spaces, Lemmas 65.45.9 and 65.40.4). It follows that the normalized blowing up is a normal (Morphisms of Spaces, Lemma 65.49.5) Nagata algebraic space. $\square$

Here is the analogue of Lemma 87.4.2 for normalized blowups.

Lemma 87.5.3. Let $X, x_ i, U_ i \to X, u_ i$ be as in (87.3.0.1) and assume $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$. Assume $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 65.49.1. Then there exists a factorization

\[ Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X \]

of $f$ where $Z_{j + 1} \to Z_ j$ is the normalized blowing up of $Z_ j$ at a closed point $z_ j$ lying over $\{ x_1, \ldots , x_ n\} $ if and only if for each $i$ there exists a factorization

\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i \]

of $g_ i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_ i$.

Proof. This follows by the exact same argument as used to prove Lemma 87.4.2. $\square$

A Nagata algebraic space is locally Noetherian.

Lemma 87.5.4. Let $S$ be a scheme. Let $X$ be a Noetherian Nagata algebraic space over $S$ with $\dim (X) = 2$. Let $f : Y \to X$ be a proper birational morphism. Then there exists a commutative diagram

\[ \xymatrix{ X_ n \ar[r] \ar[d] & X_{n - 1} \ar[r] & \ldots \ar[r] & X_1 \ar[r] & X_0 \ar[d] \\ Y \ar[rrrr] & & & & X } \]

where $X_0 \to X$ is the normalization and where $X_{i + 1} \to X_ i$ is the normalized blowing up of $X_ i$ at a closed point.

Proof. Although one can prove this lemma directly for algebraic spaces, we will continue the approach used above to reduce it to the case of schemes.

We will use that Noetherian algebraic spaces are quasi-separated and hence points have well defined residue fields (for example by Decent Spaces, Lemma 66.11.4). We will use the results of Morphisms of Spaces, Sections 65.26, 65.35, and 65.49 without further mention. We may replace $Y$ by its normalization. Let $X_0 \to X$ be the normalization. The morphism $Y \to X$ factors through $X_0$. Thus we may assume that both $X$ and $Y$ are normal.

Assume $X$ and $Y$ are normal. The morphism $f : Y \to X$ is an isomorphism over an open which contains every point of codimension $0$ and $1$ in $Y$ and every point of $Y$ over which the fibre is finite, see Spaces over Fields, Lemma 70.3.3. Hence we see that there is a finite set of closed points $T \subset |X|$ such that $f$ is an isomorphism over $X \setminus T$. By More on Morphisms of Spaces, Lemma 74.39.4 there exists an $X \setminus T$-admissible blowup $Y' \to X$ which dominates $Y$. After replacing $Y$ by the normalization of $Y'$ we see that we may assume that $Y \to X$ is representable.

Say $T = \{ x_1, \ldots , x_ r\} $. Pick elementary ├ętale neighbourhoods $(U_ i, u_ i) \to (X, x_ i)$ as in Section 87.3. For each $i$ the morphism $Y_ i = Y \times _ X U_ i \to U_ i$ is a proper birational morphism which is an isomorphism over $U_ i \setminus \{ u_ i\} $. Thus we may apply Resolution of Surfaces, Lemma 54.5.3 to find a sequence

\[ X_{i, m_ i} \to X_{i, m_ i - 1} \to \ldots \to X_1 \to X_{i, 0} = U_ i \]

of normalized blowing ups in closed points lying over $u_ i$ such that $X_{i, m_ i}$ dominates $Y_ i$. By Lemma 87.5.3 we find a sequence of normalized blowing ups

\[ X_ m \to X_{m - 1} \to \ldots \to X_1 \to X_0 = X \]

as in the statement of the lemma whose base change to our $U_ i$ produces the given sequences. It follows that $X_ m$ dominates $Y$ by the equivalence of categories of Lemma 87.3.1. $\square$


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