The general definition

We start by defining the blow up of an affine Noetherian scheme $X=Spec\,A$$X=Spec\,A$ along a closed subscheme $Z$$Z$. Since $Z$$Z$ is given as the vanishing set of some ideal $I\triangleleft A$$I\triangleleft A$ we have $Z=Spec\,A/I$$Z=Spec\,A/I$.

Definition 2.6

For a Noetherian affine scheme $X=Spec\,A$$X=Spec\,A$ and a closed subscheme $Z=Spec\,A/I$$Z=Spec\,A/I$ we define the blow up of $X$$X$ along $Z$$Z$ to be

$$\operatorname {Bl} _{Z}X=Proj\,_{A}(A\oplus I\oplus I^{2}\oplus \cdots ),$$

$$\operatorname {Bl} _{Z}X=Proj\,_{A}(A\oplus I\oplus I^{2}\oplus \cdots ),$$

where we view $A\oplus I\oplus I^{2}\oplus \cdots$$A\oplus I\oplus I^{2}\oplus \cdots$ as a graded $A$$A$-algebra whose degree $k$$k$ piece is $I^{k}$$I^{k}$ (with $I^{0}=A$$I^{0}=A$).

 

Note first that this strange-looking algebra is in fact generated in degree $1$$1$ by construction and also, by the Noetherian hypothesis, it is generated by finitely many elements, say $\{x_{1},x_{2},\ldots ,x_{s}\}$$\{x_{1},x_{2},\ldots ,x_{s}\}$. using this, we can give a more explicit model for it. The key point is to view $I$$I$ not as a subset of $A$$A$ but rather as the quotient module

$$(A\cdot X_{1}\oplus A\cdot X_{2}\oplus \cdots \oplus A\cdot X_{s})/\ker \phi ,$$

$$(A\cdot X_{1}\oplus A\cdot X_{2}\oplus \cdots \oplus A\cdot X_{s})/\ker \phi ,$$

where $X_{1},X_{2},\ldots ,X_{s}$$X_{1},X_{2},\ldots ,X_{s}$ are just formal variables and $\phi$$\phi$ is the surjective $A-$$A-$module homomorphism

$${\begin{aligned}\phi \colon A\cdot X_{1}\oplus A\cdot X_{2}\oplus \cdots \oplus A\cdot X_{s}&\longrightarrow I\\\phi \colon X_{i}&\longmapsto x_{i}.\end{aligned}}$$

$${\begin{aligned}\phi \colon A\cdot X_{1}\oplus A\cdot X_{2}\oplus \cdots \oplus A\cdot X_{s}&\longrightarrow I\\\phi \colon X_{i}&\longmapsto x_{i}.\end{aligned}}$$

In fact we can extend $\phi$$\phi$ to a surjective graded algebra homomorphism

$${\begin{aligned}\Phi \colon A[X_{1},X_{2},\ldots ,X_{s}]&\longrightarrow A\oplus I\oplus I^{2}\oplus \cdots \\\Phi \colon X_{0}^{\alpha _{0}}X_{1}^{\alpha _{1}}\ldots X_{n}^{\alpha _{n}}&\longmapsto (0,0,\ldots ,0,x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}},0,\ldots ),\end{aligned}}$$

$${\begin{aligned}\Phi \colon A[X_{1},X_{2},\ldots ,X_{s}]&\longrightarrow A\oplus I\oplus I^{2}\oplus \cdots \\\Phi \colon X_{0}^{\alpha _{0}}X_{1}^{\alpha _{1}}\ldots X_{n}^{\alpha _{n}}&\longmapsto (0,0,\ldots ,0,x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}},0,\ldots ),\end{aligned}}$$

where on the right hand side the non-zero term lies in position $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}$$\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}$ and the map is $A-$$A-$linear. In many concrete cases we have a much more concise way of writing down the kernel of this map and thus get a handle on the algebra $A\oplus I\oplus I^{2}\oplus \cdots$$A\oplus I\oplus I^{2}\oplus \cdots$. Let us see how we can compute our favourite example $\operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{2}$$\operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{2}$ with this new definition.

Example 2.2

We consider $X=Spec\,A$$X=Spec\,A$ and $Z=Spec\,A/I$$Z=Spec\,A/I$ with $A=\mathbb {C}$$A=\mathbb {C}$ and $I=(x_{1},x_{2})$$I=(x_{1},x_{2})$. So from the discussion above we have that $A\oplus I\oplus I^{2}\oplus \cdots$$A\oplus I\oplus I^{2}\oplus \cdots$ is the quotient of the graded algebra $\mathbb {C}$$\mathbb {C}$ by its ideal $(x_{1}X_{2}-X_{1}x_{2})$$(x_{1}X_{2}-X_{1}x_{2})$. So in fact we have:

$${\begin{aligned}\operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{2}&=Proj\,\left(\mathbb {C} /(x_{1}X_{2}-X_{1}x_{2})\right)\end{aligned}}$$

$${\begin{aligned}\operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{2}&=Proj\,\left(\mathbb {C} /(x_{1}X_{2}-X_{1}x_{2})\right)\end{aligned}}$$

whose closed points are precisely $\{((x_{1},x_{2}),[X_{1}:X_{2}])\colon x_{1}X_{2}-X_{1}x_{2}=0\}$$\{((x_{1},x_{2}),[X_{1}:X_{2}])\colon x_{1}X_{2}-X_{1}x_{2}=0\}$.

 

We now move on to give the general definition of a blow up. Let $X$$X$ be any Noetherian scheme and $Z$$Z$ -- a closed subscheme. Then on every affine open $U=Spec\,A$$U=Spec\,A$ in $X$$X$, $Z\cap U$$Z\cap U$ is given as the (scheme-theoretic) vanishing set of some ideal $I\triangleleft A$$I\triangleleft A$ which defines an ideal sheaf ${\tilde {I}}$${\tilde {I}}$ on $Spec\,A$$Spec\,A$. These local ideal sheaves glue together to give a sheaf $I_{Z}$$I_{Z}$. We can now consider the sheaf of graded ${\mathcal {O}}_{X}-$${\mathcal {O}}_{X}-$algebras ${\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots$${\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots$. On each affine open subset $U=Spec\,A$$U=Spec\,A$ of $X$$X$ we can then define a scheme $Proj\,\left({\mathcal {O}}_{X}(U)\oplus I_{Z}(U)\oplus I_{Z}(U)^{2}\oplus \cdots \right)$$Proj\,\left({\mathcal {O}}_{X}(U)\oplus I_{Z}(U)\oplus I_{Z}(U)^{2}\oplus \cdots \right)$. In fact these schemes glue together to define a new scheme which we denote $Proj\,_{X}\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)$$Proj\,_{X}\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)$. Moreover there is a natural morphism

$$\pi \colon Proj\,\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)\to X.$$

$$\pi \colon Proj\,\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)\to X.$$

Finally we state the most general definition of a blow up:

Definition 2.7

Let $X$$X$ be a Noetherian scheme and $Z$$Z$ a closed subscheme. Then we define the blow up of $X$$X$ along $Z$$Z$ to be

$$\operatorname {Bl} _{Z}X=Proj\,\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right).$$

$$\operatorname {Bl} _{Z}X=Proj\,\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right).$$

The exceptional divisor is $\pi ^{-1}(Z)$$\pi ^{-1}(Z)$, where $\pi \colon Proj\,_{X}\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)\to X$$\pi \colon Proj\,_{X}\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)\to X$ is the natural morphism.