## 38.31 Applications

In this section we apply some of the results above.

Lemma 38.31.1. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume

1. $X \to S$ is of finite type and quasi-separated, and

2. $X_ U \to U$ is flat and locally of finite presentation.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is flat and of finite presentation over $S'$.

Proof. Since $X \to S$ is quasi-compact and quasi-separated by assumption, the strict transform of $X$ with respect to a blowing up $S' \to S$ is also quasi-compact and quasi-separated. Hence to prove the lemma it suffices to find a $U$-admissible blowup such that the strict transform is flat and locally of finite presentation. Let $X = W_1 \cup \ldots \cup W_ n$ be a finite affine open covering. If we can find a $U$-admissible blowup $S_ i \to S$ such that the strict transform of $W_ i$ is flat and locally of finite presentation, then there exists a $U$-admissible blowing up $S' \to S$ dominating all $S_ i \to S$ which does the job (see Divisors, Lemma 31.34.4; see also Remark 38.30.1). Hence we may assume $X$ is affine.

Assume $X$ is affine. By Morphisms, Lemma 29.39.2 we can choose an immersion $j : X \to \mathbf{A}^ n_ S$ over $S$. Let $V \subset \mathbf{A}^ n_ S$ be a quasi-compact open subscheme such that $j$ induces a closed immersion $i : X \to V$ over $S$. Apply Theorem 38.30.7 to $V \to S$ and the quasi-coherent module $i_*\mathcal{O}_ X$ to obtain a $U$-admissible blowup $S' \to S$ such that the strict transform of $i_*\mathcal{O}_ X$ is flat over $S'$ and of finite presentation over $\mathcal{O}_{V \times _ S S'}$. Let $X'$ be the strict transform of $X$ with respect to $S' \to S$. Let $i' : X' \to V \times _ S S'$ be the induced morphism. Since taking strict transform commutes with pushforward along affine morphisms (Divisors, Lemma 31.33.4), we see that $i'_*\mathcal{O}_{X'}$ is flat over $S$ and of finite presentation as a $\mathcal{O}_{V \times _ S S'}$-module. This implies the lemma. $\square$

Lemma 38.31.2. Let $S$ be a quasi-compact and quasi-separated scheme. Let $X$ be a scheme over $S$. Let $U \subset S$ be a quasi-compact open. Assume

1. $X \to S$ is proper, and

2. $X_ U \to U$ is finite locally free.

Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform of $X$ is finite locally free over $S'$.

Proof. By Lemma 38.31.1 we may assume that $X \to S$ is flat and of finite presentation. After replacing $S$ by a $U$-admissible blowup if necessary, we may assume that $U \subset S$ is scheme theoretically dense. Then $f$ is finite by Lemma 38.11.4. Hence $f$ is finite locally free by Morphisms, Lemma 29.48.2. $\square$

Lemma 38.31.3. Let $\varphi : X \to S$ be a separated morphism of finite type with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi ^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is isomorphic to an open subscheme of $S'$.

Proof. The discussion in Remark 38.30.1 applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Next, we do another $U$-admissible blowup to get to the situation where $X \to S$ is flat and of finite presentation, see Lemma 38.31.1. In this case the result follows from Lemma 38.11.5. $\square$

The following lemma says that a proper modification can be dominated by a blowup.

Lemma 38.31.4. Let $\varphi : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Let $U \subset S$ be a quasi-compact open such that $\varphi ^{-1}U \to U$ is an isomorphism. Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$, i.e., such that there exists a factorization $S' \to X \to S$ of the blowup morphism.

Proof. The discussion in Remark 38.30.1 applies. Thus we may do a first $U$-admissible blowup and assume the complement $S \setminus U$ is the support of an effective Cartier divisor $D$. In particular $U$ is scheme theoretically dense in $S$. Choose another $U$-admissible blowup $S' \to S$ such that the strict transform $X'$ of $X$ is an open subscheme of $S'$, see Lemma 38.31.3. Since $X' \to S'$ is proper, and $U \subset S'$ is dense, we see that $X' = S'$. Some details omitted. $\square$

Lemma 38.31.5. Let $S$ be a scheme. Let $U \subset W \subset S$ be open subschemes. Let $f : X \to W$ be a morphism and let $s : U \to X$ be a morphism such that $f \circ s = \text{id}_ U$. Assume

1. $f$ is proper,

2. $S$ is quasi-compact and quasi-separated, and

3. $U$ and $W$ are quasi-compact.

Then there exists a $U$-admissible blowup $b : S' \to S$ and a morphism $s' : b^{-1}(W) \to X$ extending $s$ with $f \circ s' = b|_{b^{-1}(W)}$.

Proof. We may and do replace $X$ by the scheme theoretic image of $s$. Then $X \to W$ is an isomorphism over $U$, see Morphisms, Lemma 29.6.8. By Lemma 38.31.4 there exists a $U$-admissible blowup $W' \to W$ and an extension $W' \to X$ of $s$. We finish the proof by applying Divisors, Lemma 31.34.3 to extend $W' \to W$ to a $U$-admissible blowup of $S$. $\square$

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