Let me take $X:=x^2-z^3y^3=0$ to be our surface over $\mathbb{A}^3$. Then we see we have a singular locus on the y and z axis. So, I will blow up initial on the Z-axis $V = Z(<x,y>)$.
Then $\Gamma_{X,Z} = \{ (x,y,z)[u:v] \mid \underline{x} \in X-Z \}$ with relation $xv = uy$and we want to find the closure of this in $ \mathbb{A}^3 \times \mathbb{P}^1$.
So, we look for the affine charts of this closure. Take $u=1$ , then we get $x^2(1-v^3z^3)$, where we don't need account for loci $x=0$. Hence, $1 -xv^33z^3 \subset \mathbb{A}^3$ will be closure of the affine space and is non singular.
However, for the chart $v=1$, this is clearly affine chart of our blow up will correspond to $u^2 -yz^3 \subset \mathbb{A}^3$, which is singular.
This I will now be blowing up this affine chart. However, I am starting to get slightly, lost in terms of what is my space looking like. Also, are my steps justified. I.e do I need to explicitly need to describe by blow up as a variety, by gluing the charts together.
