The Stacks project

31.32 Blowing up

Blowing up is an important tool in algebraic geometry.

Definition 31.32.1. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subscheme corresponding to $\mathcal{I}$, see Schemes, Definition 26.10.2. 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” scheme 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 Constructions, Lemma 27.16.11. Note that $\mathcal{O}_{X'}(1)$ is $b$-relatively very ample, even though $b$ need not be of finite type or even quasi-compact, because $X'$ comes equipped with a closed immersion into $\mathbf{P}(\mathcal{I})$, see Morphisms, Example 29.38.3.

Lemma 31.32.2. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open subscheme of $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_ U$. If $b : X' \to X$ is the blowup of $X$ in $\mathcal{I}$, then there is a canonical isomorphism

\[ b^{-1}(U) = \text{Proj}(\bigoplus \nolimits _{d \geq 0} I^ d) \]

of $b^{-1}(U)$ with the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $b^{-1}(U)$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$.

Proof. The first statement is clear from the construction of the relative Proj via glueing, see Constructions, Section 27.15. For $a \in I$ denote $a^{(1)}$ the element $a$ seen as an element of degree $1$ in the Rees algebra $\bigoplus _{n \geq 0} I^ n$. Since these elements generate the Rees algebra over $A$ we see that $\text{Proj}(\bigoplus _{d \geq 0} I^ d)$ is covered by the affine opens $D_{+}(a^{(1)})$. The affine scheme $D_{+}(a^{(1)})$ is the spectrum of the affine blowup algebra $A' = A[\frac{I}{a}]$, see Algebra, Definition 10.70.1. This finishes the proof. $\square$

slogan

Lemma 31.32.3. Let $X_1 \to X_2$ be a flat morphism of schemes. Let $Z_2 \subset X_2$ be a closed subscheme. 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 schemes.

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 $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (by definition of the inverse image, see Schemes, Definition 26.17.7). By Constructions, Lemma 27.16.10 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 31.32.4. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. 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. As blowing up commutes with restrictions to open subschemes (Lemma 31.32.3) the first statement just means that $X' = X$ if $Z = \emptyset $. In this case we are blowing up in the ideal sheaf $\mathcal{I} = \mathcal{O}_ X$ and the result follows from Constructions, Example 27.8.14.

The second statement is local on $X$, hence we may assume $X$ affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. By Lemma 31.32.2 we see that $X'$ is covered by the spectra of the affine blowup algebras $A' = A[\frac{I}{a}]$. Then $IA' = aA'$ and $a$ maps to a nonzerodivisor in $A'$ according to Algebra, Lemma 10.70.2. This proves the lemma as the inverse image of $Z$ in $\mathop{\mathrm{Spec}}(A')$ corresponds to $\mathop{\mathrm{Spec}}(A'/IA') \subset \mathop{\mathrm{Spec}}(A')$.

Consider the canonical map $\psi _{univ, 1} : b^*\mathcal{I} \to \mathcal{O}_{X'}(1)$, see discussion following Constructions, Definition 27.16.7. We claim that this factors through an isomorphism $\mathcal{I}_ E \to \mathcal{O}_{X'}(1)$ (which proves the final assertion). Namely, on the affine open corresponding to the blowup algebra $A' = A[\frac{I}{a}]$ mentioned above $\psi _{univ, 1}$ corresponds to the $A'$-module map

\[ I \otimes _ A A' \longrightarrow \left(\Big(\bigoplus \nolimits _{d \geq 0} I^ d\Big)_{a^{(1)}}\right)_1 \]

where $a^{(1)}$ is as in Algebra, Definition 10.70.1. We omit the verification that this is the map $I \otimes _ A A' \to IA' = aA'$. $\square$

Lemma 31.32.5 (Universal property blowing up). Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{C}$ be the full subcategory of $(\mathit{Sch}/X)$ 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 31.32.4. 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 the material in Constructions, Section 27.16 the triple $(1, f : Y \to X, \psi )$ 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 31.13.4. Thus the morphism $Y \to X'$ is unique by Morphisms, Lemma 29.7.10 (also $b$ is separated by Constructions, Lemma 27.16.9). $\square$

Lemma 31.32.6. Let $b : X' \to X$ be the blowing up of the scheme $X$ along a closed subscheme $Z$. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$ and let $I \subset A$ be the ideal corresponding to $Z \cap U$. Let $a \in I$ and let $x' \in X'$ be a point mapping to a point of $U$. Then $x'$ is a point of the affine open $U' = \mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ if and only if the image of $a$ in $\mathcal{O}_{X', x'}$ cuts out the exceptional divisor.

Proof. Since the exceptional divisor over $U'$ is cut out by the image of $a$ in $A' = A[\frac{I}{a}]$ one direction is clear. Conversely, assume that the image of $a$ in $\mathcal{O}_{X', x'}$ cuts out $E$. Since every element of $I$ maps to an element of the ideal defining $E$ over $b^{-1}(U)$ we see that elements of $I$ become divisible by $a$ in $\mathcal{O}_{X', x'}$. Thus for $f \in I^ n$ we can write $f = \psi (f) a^ n$ for some $\psi (f) \in \mathcal{O}_{X', x'}$. Observe that since $a$ maps to a nonzerodivisor of $\mathcal{O}_{X', x'}$ the element $\psi (f)$ is uniquely characterized by this. Then we define

\[ A' \longrightarrow \mathcal{O}_{X', x'},\quad f/a^ n \longmapsto \psi (f) \]

Here we use the description of blowup algebras given following Algebra, Definition 31.32.1. The uniqueness mentioned above shows that this is an $A$-algebra homomorphism. This gives a morphism $\mathop{\mathrm{Spec}}(\mathcal{O}_{X', x"}) \to \mathop{\mathrm{Spec}}(A') = U'$. By the universal property of blowing up (Lemma 31.32.5) this is a morphism over $X'$, which of course implies that $x' \in U'$. $\square$

Lemma 31.32.7. Let $X$ be a scheme. 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 31.32.5). $\square$

Lemma 31.32.8. Let $X$ be a scheme. 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.

Lemma 31.32.9. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a nonzero quasi-coherent sheaf of ideals. If $X$ is integral, then the blowup $X'$ of $X$ in $\mathcal{I}$ is integral.

Lemma 31.32.10. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $b : X' \to X$ be the blowing up of $X$ along $Z$. Then $b$ induces an bijective map from the set of generic points of irreducible components of $X'$ to the set of generic points of irreducible components of $X$ which are not in $Z$.

Proof. The exceptional divisor $E \subset X'$ is an effective Cartier divisor (Lemma 31.32.4) hence is nowhere dense in $X'$ (Lemma 31.13.4). On the other hand, $X' \setminus E \to X \setminus Z$ is an isomorphism. The lemma follows. $\square$

Lemma 31.32.11. Let $X$ be a scheme. Let $b : X' \to X$ be a blowup of $X$ in a closed subscheme. The pullback $b^{-1}D$ is defined for all effective Cartier divisors $D \subset X$ and pullbacks of meromorphic functions are defined for $b$ (Definitions 31.13.12 and 31.23.4).

Proof. By Lemmas 31.32.2 and 31.13.2 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 31.32.12. Let $X$ be a scheme. Let $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X$ 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 31.32.4. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma 31.32.11. 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 31.32.5 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 31.13.9. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma 31.32.5. 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 31.32.13. Let $X$ be a scheme. 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

  1. $b : X' \to X$ is a projective morphism, and

  2. $\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf.

Proof. The surjection of graded $\mathcal{O}_ X$-algebras

\[ \text{Sym}_{\mathcal{O}_ X}^*(\mathcal{I}) \longrightarrow \bigoplus \nolimits _{d \geq 0} \mathcal{I}^ d \]

defines via Constructions, Lemma 27.18.5 a closed immersion

\[ X' = \underline{\text{Proj}}_ X (\bigoplus \nolimits _{d \geq 0} \mathcal{I}^ d) \longrightarrow \mathbf{P}(\mathcal{I}). \]

Hence $b$ is projective, see Morphisms, Definition 29.43.1. The second statement follows for example from the characterization of relatively ample invertible sheaves in Morphisms, Lemma 29.37.4. Some details omitted. $\square$

slogan

Lemma 31.32.14. Let $X$ be a quasi-compact and quasi-separated scheme. Let $Z \subset X$ be a closed subscheme of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subscheme of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subscheme $Y \subset X$ of finite presentation, such that

  1. $Y = Z \cup b(Z')$ set theoretically, 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, Lemma 29.21.7. 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 of $X$ by Properties, Lemma 28.24.1. Since $b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism (Lemma 31.32.4) the same result shows that $b^{-1}(X \setminus Z) \setminus Z'$ is quasi-compact open in $X'$. Hence $U = X \setminus (Z \cup b(Z'))$ is quasi-compact open in $X$. By Lemma 31.31.5 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 $V(\mathcal{I}\mathcal{J}) = Z \cup b(Z')$ set theoretically. 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 31.32.12. 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 31.32.7 and 31.32.12.

To see the displayed equality of the ideals we may work locally. With notation $A$, $I$, $a \in I$ as in Lemma 31.32.2 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 (7)

Comment #1610 by Antoine Chambert-Loir on

Question of terminology: The text uses blowing up, blow-up, blow up and blowup. Why not sticking to one? (One could also favor blowing up for the morphism and blow up for the scheme.) An related important question is that of their plurals.

Comment #1668 by on

OK, this is a bit of a big job. I personally like using all manner of different words for the same thing, but I understand this is not always a good idea. Maybe we should have a vote on the blog about which is the best. Stay tuned!

Comment #1842 by LauBru on

I think that it is missing an important fact about blow ups:

Let be a ground scheme and a closed embedding of -schemes. If is flat over then the blow up of X in Y commute with any base change .

I was searching this result for month, finally I proved it by hand, but surly this result can be useful. Well, I hope this help you.

Comment #1879 by on

Dear LauBru, this is not true: a counter example is to take as base the spectrum of the ring where is a field and is a variable, to take as the spectrum of and to take as the spectrum of the quotient ring . Finally, take the spectrum of .

For everybody reading and leaving comments: of course the results in a particular section do not cover all possible results about the topic discussed in that section. Namely, we try to have a skeleton outline of the basic material and then come back to it with more as needed in development of later material. Moreover, sometimes a result cannot be formulated or proved immediately because it needs more terminology or results proven later in the Stacks project.

Comment #2611 by Dario Weißmann on

Typo in the proof of 30.29.6: "one direct is clear" should be "one direction is clear"

Comment #2823 by Antoine Chambert-Loir on

To add some noise to the (non-existing) discussion about comments 1610 and 1668, I note that the term blow up appears in various guises: blow-up, blow up, and blowup (not mentioning plurals).

Comment #2924 by on

Dear Antoine Chambert-Loir: that blows! Tried to improve this here.


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