The Stacks project

68.17 Blowing up

Blowing up is an important tool in algebraic geometry.

Definition 68.17.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subspace corresponding to $\mathcal{I}$ (Morphisms of Spaces, Lemma 64.13.1). The blowing up of $X$ along $Z$, or the blowing up of $X$ in the ideal sheaf $\mathcal{I}$ is the morphism

\[ b : \underline{\text{Proj}}_ X \left(\bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n\right) \longrightarrow X \]

The exceptional divisor of the blowup is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the center of the blowup.

We will see later that the exceptional divisor is an effective Cartier divisor. Moreover, the blowing up is characterized as the “smallest” algebraic space over $X$ such that the inverse image of $Z$ is an effective Cartier divisor.

If $b : X' \to X$ is the blowup of $X$ in $Z$, then we often denote $\mathcal{O}_{X'}(n)$ the twists of the structure sheaf. Note that these are invertible $\mathcal{O}_{X'}$-modules and that $\mathcal{O}_{X'}(n) = \mathcal{O}_{X'}(1)^{\otimes n}$ because $X'$ is the relative Proj of a quasi-coherent graded $\mathcal{O}_ X$-algebra which is generated in degree $1$, see Lemma 68.11.11.

Lemma 68.17.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme étale over $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_ U$. If $X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism

\[ U \times _ X X' = \text{Proj}(\bigoplus \nolimits _{d \geq 0} I^ d) \]

of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \times _ X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.

Proof. Note that the restriction $\mathcal{I}|_ U$ is equal to the pullback of $\mathcal{I}$ via the morphism $U \to X$, see Properties of Spaces, Section 63.26. Thus the lemma follows on combining Lemma 68.11.2 with Divisors, Lemma 31.32.2. $\square$

Lemma 68.17.3. Let $S$ be a scheme. Let $X_1 \to X_2$ be a flat morphism of algebraic spaces over $S$. Let $Z_2 \subset X_2$ be a closed subspace. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_ i$ be the blowup of $Z_ i$ in $X_ i$. Then there exists a cartesian diagram

\[ \xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 } \]

of algebraic spaces over $S$.

Proof. Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (see Morphisms of Spaces, Definition 64.13.2 and discussion following the definition). By Lemma 68.11.5 we see that $X_1 \times _{X_2} X_2'$ is the relative Proj of $\bigoplus _{n \geq 0} g^*\mathcal{I}_2^ n$. Because $g$ is flat the map $g^*\mathcal{I}_2^ n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^ n$. Thus we see that $X_1 \times _{X_2} X_2' = X_1'$. $\square$

Lemma 68.17.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties:

  1. $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism,

  2. the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$,

  3. there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 68.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.4. $\square$

slogan

Lemma 68.17.5 (Universal property blowing up). Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. Let $\mathcal{C}$ be the full subcategory of $(\textit{Spaces}/X)$ consisting of $Y \to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$ is a final object of $\mathcal{C}$.

Proof. We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to Lemma 68.17.4. Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists a unique morphism $Y \to X'$ over $X$. Let $D = f^{-1}(Z)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of $Z$ and let $\mathcal{I}_ D$ be the ideal sheaf of $D$. Then $f^*\mathcal{I} \to \mathcal{I}_ D$ is a surjection to an invertible $\mathcal{O}_ Y$-module. This extends to a map $\psi : \bigoplus f^*\mathcal{I}^ d \to \bigoplus \mathcal{I}_ D^ d$ of graded $\mathcal{O}_ Y$-algebras. (We observe that $\mathcal{I}_ D^ d = \mathcal{I}_ D^{\otimes d}$ as $D$ is an effective Cartier divisor.) By Lemma 68.11.11. the triple $(f : Y \to X, \mathcal{I}_ D, \psi )$ defines a morphism $Y \to X'$ over $X$. The restriction

\[ Y \setminus D \longrightarrow X' \setminus b^{-1}(Z) = X \setminus Z \]

is unique. The open $Y \setminus D$ is scheme theoretically dense in $Y$ according to Lemma 68.6.4. Thus the morphism $Y \to X'$ is unique by Morphisms of Spaces, Lemma 64.17.8 (also $b$ is separated by Lemma 68.11.6). $\square$

Lemma 68.17.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$.

Proof. Immediate from the universal property of blowups (Lemma 68.17.5). $\square$

Lemma 68.17.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\mathcal{I}$ is reduced.

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 68.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.8. $\square$

Lemma 68.17.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be the blowup of $X$ in a closed subspace. If $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 64.49.1 then so does $X'$.

Proof. Follows immediately from the lemma cited in the statement, the étale local description of blowing ups in Lemma 68.17.2, and Divisors, Lemma 31.32.10. $\square$

Lemma 68.17.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be a blowup of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^{-1}D$ is defined (see Definition 68.6.10).

Proof. By Lemmas 68.17.2 and 68.6.2 this reduces to the following algebra fact: Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$ a nonzerodivisor. Then the image of $x$ in $A[\frac{I}{a}]$ is a nonzerodivisor. Namely, suppose that $x (y/a^ n) = 0$ in $A[\frac{I}{a}]$. Then $a^ mxy = 0$ in $A$ for some $m$. Hence $a^ my = 0$ as $x$ is a nonzerodivisor. Whence $y/a^ n$ is zero in $A[\frac{I}{a}]$ as desired. $\square$

Lemma 68.17.10. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ and $\mathcal{J}$ be quasi-coherent sheaves of ideals. Let $b : X' \to X$ be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$ is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$.

Proof. Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective Cartier divisor by Lemma 68.17.4. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma 68.17.9. Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective Cartier divisor). Consider the effective Cartier divisor $E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is $(b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}$. Hence according to Lemma 68.17.5 there is a canonical morphism from $X''$ to the blowup $c : Y \to X$ of $X$ in $\mathcal{I}\mathcal{J}$. Conversely, as $\mathcal{I}\mathcal{J}$ pulls back to an invertible ideal we see that $c^{-1}\mathcal{I}\mathcal{O}_ Y$ defines an effective Cartier divisor, see Lemma 68.6.8. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma 68.17.5. Then $(c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_ Y = c^{-1}\mathcal{J}\mathcal{O}_ Y$ which also defines an effective Cartier divisor. Thus a morphism $c'' : Y \to X''$ over $X'$. We omit the verification that this morphism is inverse to the morphism $X'' \to Y$ constructed earlier. $\square$

Lemma 68.17.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then $b : X' \to X$ is a proper morphism.

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 68.17.3) we can prove each of these statements after base change to $U$ (see Morphisms of Spaces, Lemma 64.40.2). This reduces us to the case of schemes. In this case the morphism $b$ is projective by Divisors, Lemma 31.32.13 hence proper by Morphisms, Lemma 29.41.5. $\square$

Lemma 68.17.12. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Let $Z \subset X$ be a closed subspace of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subspace of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subspace $Y \subset X$ of finite presentation, such that

  1. $|Y| = |Z| \cup |b|(|Z'|)$, and

  2. the composition $X'' \to X$ is isomorphic to the blowing up of $X$ in $Y$.

Proof. The condition that $Z \to X$ is of finite presentation means that $Z$ is cut out by a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, see Morphisms of Spaces, Lemma 64.28.12. Write $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$ so that $X' = \underline{\text{Proj}}(\mathcal{A})$. Note that $X \setminus Z$ is a quasi-compact open subspace of $X$ by Limits of Spaces, Lemma 67.14.1. Since $b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism (Lemma 68.17.4) the same result shows that $b^{-1}(X \setminus Z) \setminus Z'$ is quasi-compact open subspace in $X'$. Hence $U = X \setminus (Z \cup b(Z'))$ is quasi-compact open subspace in $X$. By Lemma 68.16.3 there exist a $d > 0$ and a finite type $\mathcal{O}_ X$-submodule $\mathcal{F} \subset \mathcal{I}^ d$ such that $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ and such that the support of $\mathcal{I}^ d/\mathcal{F}$ is contained in $X \setminus U$.

Since $\mathcal{F} \subset \mathcal{I}^ d$ is an $\mathcal{O}_ X$-submodule we may think of $\mathcal{F} \subset \mathcal{I}^ d \subset \mathcal{O}_ X$ as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this $\mathcal{J} \subset \mathcal{O}_ X$ to prevent confusion. Since $\mathcal{I}^ d / \mathcal{J}$ and $\mathcal{O}/\mathcal{I}^ d$ are supported on $|X| \setminus |U|$ we see that $|V(\mathcal{J})|$ is contained in $|X| \setminus |U|$. Conversely, as $\mathcal{J} \subset \mathcal{I}^ d$ we see that $|Z| \subset |V(\mathcal{J})|$. Over $X \setminus Z \cong X' \setminus b^{-1}(Z)$ the sheaf of ideals $\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence $|V(\mathcal{J})|$ equals $|Z| \cup |b|(|Z'|)$. It follows that also $|V(\mathcal{I}\mathcal{J})| = |Z| \cup |b|(|Z'|)$. Moreover, $\mathcal{I}\mathcal{J}$ is an ideal of finite type as a product of two such. We claim that $X'' \to X$ is isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$ which finishes the proof of the lemma by setting $Y = V(\mathcal{I}\mathcal{J})$.

First, recall that the blowup of $X$ in $\mathcal{I}\mathcal{J}$ is the same as the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$, see Lemma 68.17.10. Hence it suffices to show that the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$ agrees with the blowup of $X'$ in $Z'$. We will show that

\[ b^{-1}\mathcal{J} \mathcal{O}_{X'} = \mathcal{I}_ E^ d \mathcal{I}_{Z'} \]

as ideal sheaves on $X''$. This will prove what we want as $\mathcal{I}_ E^ d$ cuts out the effective Cartier divisor $dE$ and we can use Lemmas 68.17.6 and 68.17.10.

To see the displayed equality of the ideals we may work locally. With notation $A$, $I$, $a \in I$ as in Lemma 68.17.2 we see that $\mathcal{F}$ corresponds to an $R$-submodule $M \subset I^ d$ mapping isomorphically to an ideal $J \subset R$. The condition $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ means that $Z' \cap \mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ is cut out by the ideal generated by the elements $m/a^ d$, $m \in M$. Say the element $m \in M$ corresponds to the function $f \in J$. Then in the affine blowup algebra $A' = A[\frac{I}{a}]$ we see that $f = (a^ dm)/a^ d = a^ d (m/a^ d)$. Thus the equality holds. $\square$


Comments (2)

Comment #2252 by Federico Scavia on

Minor correction: in the first definition, I think should be a subspace, not a subscheme.


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