# The general definition

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

Definition 2.6

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

${\displaystyle \operatorname {Bl} _{Z}X=Proj\,_{A}(A\oplus I\oplus I^{2}\oplus \cdots ),}$
where we view ${\displaystyle A\oplus I\oplus I^{2}\oplus \cdots }$ as a graded ${\displaystyle A}$-algebra whose degree ${\displaystyle k}$ piece is ${\displaystyle I^{k}}$ (with ${\displaystyle I^{0}=A}$).

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

${\displaystyle (A\cdot X_{1}\oplus A\cdot X_{2}\oplus \cdots \oplus A\cdot X_{s})/\ker \phi ,}$
where ${\displaystyle X_{1},X_{2},\ldots ,X_{s}}$ are just formal variables and ${\displaystyle \phi }$ is the surjective ${\displaystyle A-}$module homomorphism
{\displaystyle {\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 ${\displaystyle \phi }$ to a surjective graded algebra homomorphism
{\displaystyle {\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 ${\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}$ and the map is ${\displaystyle A-}$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 ${\displaystyle A\oplus I\oplus I^{2}\oplus \cdots }$. Let us see how we can compute our favourite example ${\displaystyle \operatorname {Bl} _{\mathbf {0} }\mathbb {C} ^{2}}$ with this new definition.

Example 2.2

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

{\displaystyle {\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 ${\displaystyle \{((x_{1},x_{2}),[X_{1}:X_{2}])\colon x_{1}X_{2}-X_{1}x_{2}=0\}}$.

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

${\displaystyle \pi \colon Proj\,\left({\mathcal {O}}_{X}\oplus I_{Z}\oplus I_{Z}^{2}\oplus \cdots \right)\to X.}$
Finally we state the most general definition of a blow up:

Definition 2.7

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

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