As described in these notes, I am trying to compute the blow up of $\mathbb C^2=\text{ Maxspec }\mathbb C[x,y]$ along the subvariety corresponding to the ideal $\langle\ x^2,y\ \rangle$ but I am having trouble with the steps.
Now the blow up is given by the closure of the image of the map $f$ defined as follows - $$f:\mathbb C^2\setminus\mathcal Z(x^2,y)\to \mathbb C^2\times \mathbb P^1\quad\text{is given by}\quad (a,b)\mapsto(a,b,[a^2:b])$$
Now if $[x_0:x_1]$ are the homogeneous coordinates of $\mathbb P^1$ then $\mathbb {C^2\times P^1}$ is covered by $\mathbb C^2\times U_0$ and $\mathbb C^2\times U_1$ where $U_i=\{\ [a_0:a_1]\ |\ a_i\neq0\ \}$.
Let $A_0=\overline{\text{ Im }f}\cap(\mathbb C^2\times U_0)$ and $A_1=\overline{\text{ Im }f}\cap(\mathbb C^2\times U_1)$ then the blow up is obtained by appropriately glueing $A_0$ and $A_1$. However I am unable to explicitely write this "appropriate glueing" without a proper description of $A_0$ and $A_1$. By that I mean I want to write the $A_i$ as the vanishing of some ideals in $\mathbb {C^2\times P^1}$. This way I can look at glueing maps at the coordinate ring level.
Could someone hepl me describe $A_0$ and $A_1$?
Thank you.
In more generality, there's the following result: Let $X \subset \mathbb{A}^n$ be an affine variety and $Z\subset X$ be a closed smooth subvariety, which is defined by the vanishing of the polynomials ${f_1, ..., f_k}$ in $\mathbb{A}^n$. Then the blow-up with respect to the subvariety $Z$, denoted $Bl_Z\mathbb{A}^n$, in $\mathbb{A}^n \times {\mathbb{P}}^{k-1}$ (note the "degree of ${\mathbb{P}}$" corresponds to the number of defining polynomials of $Z$) is given by $\{ y_if_j=y_jf_i \vert i,j=1,...,k \}$.
So in your case, the blow-up is the set $\{ (x,y),[a_0,a_1] \in \mathbb{C}^2 \times \mathbb{P}^1 \text{ } \vert \text{ }a_0f_2(x,y)=a_1f_1(x,y) \}$ where $f_1(x,y)=x^2$ and $f_2(x,y)=y$. So your $A_0$ simply becomes $A_0=\{(x,y),[1:a_1]\text{ } \vert \text{ } y=a_1x^2 \}=\{(x,a_1x^2),[1:a_1]\} \cong \mathbb{C}^2_{x,a_1}$. Similarly with $A_1$.
By the way, do you notice what subvariety you are blowing up?