The Stacks project

31.34 Admissible blowups

To have a bit more control over our blowups we introduce the following standard terminology.

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 $Z$.

We recall that $Z \to X$ is of finite presentation if and only if the ideal sheaf $\mathcal{I}_ Z \subset \mathcal{O}_ X$ is of finite type, see Morphisms, Lemma 29.21.7. In particular, a $U$-admissible blowup is a projective morphism, see Lemma 31.32.13. Note that there can be multiple centers which give rise to the same morphism. Hence the requirement is just the existence of some center disjoint from $U$ which produces $X'$. Finally, as the morphism $b : X' \to X$ is an isomorphism over $U$ (see Lemma 31.32.4) we will often abuse notation and think of $U$ as an open subscheme of $X'$ as well.


Lemma 31.34.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. Let $b : X' \to X$ be a $U$-admissible blowup. Let $X'' \to X'$ be a $U$-admissible blowup. Then the composition $X'' \to X$ is a $U$-admissible blowup.

Proof. Immediate from the more precise Lemma 31.32.14. $\square$

Lemma 31.34.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V \subset X$ be quasi-compact open subschemes. Let $b : V' \to V$ be a $U \cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \to X$ whose restriction to $V$ is $V'$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_ U$ and whose restriction to $V$ is $\mathcal{I}$ (see Sheaves, Section 6.33). By Properties, Lemma 28.22.2 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows. $\square$

Lemma 31.34.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. Let $b_ i : X_ i \to X$, $i = 1, \ldots , n$ be $U$-admissible blowups. There exists a $U$-admissible blowup $b : X' \to X$ such that (a) $b$ factors as $X' \to X_ i \to X$ for $i = 1, \ldots , n$ and (b) each of the morphisms $X' \to X_ i$ is a $U$-admissible blowup.

Proof. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_ i)$ is disjoint from $U$ and such that $X_ i$ is isomorphic to the blowup of $X$ in $\mathcal{I}_ i$. Set $\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_ n$ and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then $X' \to X$ factors through $b_ i$ by Lemma 31.32.12. $\square$


Lemma 31.34.5. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V$ be quasi-compact disjoint open subschemes of $X$. Then there exist a $U \cup V$-admissible blowup $b : X' \to X$ such that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ with $b^{-1}(U) \subset X'_1$ and $b^{-1}(V) \subset X'_2$.

Proof. Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp. $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp. $X \setminus V = V(\mathcal{J})$, see Properties, Lemma 28.24.1. Then $V(\mathcal{I}\mathcal{J}) = X$ set theoretically, hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^ n \mathcal{J}^ n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^ n$ and $\mathcal{J}$ by $\mathcal{J}^ n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $V(\mathcal{I} + \mathcal{J}) = X \setminus U \cup V$. We will show that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ such that $b^{-1}\mathcal{I}|_{X'_2} = 0$ and $b^{-1}\mathcal{J}|_{X'_1} = 0$ which will prove the lemma.

We will use the description of the blowing up in Lemma 31.32.2. Suppose that $U = \mathop{\mathrm{Spec}}(A) \subset X$ is an affine open such that $\mathcal{I}|_ U$, resp. $\mathcal{J}|_ U$ corresponds to the finitely generated ideal $I \subset A$, resp. $J \subset A$. Then

\[ b^{-1}(U) = \text{Proj}(A \oplus (I + J) \oplus (I + J)^2 \oplus \ldots ) \]

This is covered by the affine open subsets $A[\frac{I + J}{x}]$ and $A[\frac{I + J}{y}]$ with $x \in I$ and $y \in J$. Since $x \in I$ is a nonzerodivisor in $A[\frac{I + J}{x}]$ and $IJ = 0$ we see that $J A[\frac{I + J}{x}] = 0$. Since $y \in J$ is a nonzerodivisor in $A[\frac{I + J}{y}]$ and $IJ = 0$ we see that $I A[\frac{I + J}{y}] = 0$. Moreover,

\[ \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{x}}]) \cap \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{y}}]) = \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{xy}}]) = \emptyset \]

because $xy$ is both a nonzerodivisor and zero. Thus $b^{-1}(U)$ is the disjoint union of the open subscheme $U_1$ defined as the union of the standard opens $\mathop{\mathrm{Spec}}(A[\frac{I + J}{x}])$ for $x \in I$ and the open subscheme $U_2$ which is the union of the affine opens $\mathop{\mathrm{Spec}}(A[\frac{I + J}{y}])$ for $y \in J$. We have seen that $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_2$ and $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_1$. We omit the verification that these open subschemes glue to global open subschemes $X'_1$ and $X'_2$. $\square$

Lemma 31.34.6. Let $X$ be a locally Noetherian scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a regular meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. The blowup $b : X' \to X$ in the ideal of denominators of $s$ is $U$-admissible. There exists an effective Cartier divisor $D \subset X'$ and an isomorphism

\[ b^*\mathcal{L} = \mathcal{O}_{X'}(D - E), \]

where $E \subset X'$ is the exceptional divisor such that the meromorphic section $b^*s$ corresponds, via the isomorphism, to the meromorphic section $1_ D \otimes (1_ E)^{-1}$.

Proof. From the definition of the ideal of denominators in Definition 31.23.10 we immediately see that $b$ is a $U$-admissible blowup. For the notation $1_ D$, $1_ E$, and $\mathcal{O}_{X'}(D - E)$ please see Definition 31.14.1. The pullback $b^*s$ is defined by Lemmas 31.32.11 and 31.23.8. Thus the statement of the lemma makes sense. We can reinterpret the final assertion as saying that $b^*s$ is a global regular section of $b^*\mathcal{L}(E)$ whose zero scheme is $D$. This uniquely defines $D$ hence to prove the lemma we may work affine locally on $X$ and $X'$. Assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{L} = \mathcal{O}_ X$. Then $s$ is a regular meromorphic function and shrinking further we may assume $s = a'/a$ with $a', a \in A$ nonzerodivisors. Then the ideal of denominators of $s$ corresponds to the ideal $I = \{ x \in A \mid xa' \in aA\} $. Recall that $X'$ is covered by spectra of affine blowup algebras $A' = A[\frac{I}{x}]$ with $x \in I$ (Lemma 31.32.2). Fix $x \in I$ and write $xa' = a a''$ for some $a'' \in A$. The divisor $E \subset X'$ is cut out by $x \in A'$ over the spectrum of $A'$ and hence $1/x$ is a generator of $\mathcal{O}_{X'}(E)$ over $\mathop{\mathrm{Spec}}(A')$. Finally, in the total quotient ring of $A'$ we have $a'/a = a''/x$. Hence $b^*s = a'/a$ restricts to a regular section of $\mathcal{O}_{X'}(E)$ which is over $\mathop{\mathrm{Spec}}(A')$ given by $a''/x$. This finishes the proof. (The divisor $D \cap \mathop{\mathrm{Spec}}(A')$ is cut out by the image of $a''$ in $A'$.) $\square$

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