Understanding BlowUp Computation in Singular

Question

Many of us might know that "Singular" is a computer algebra system for Algebraic Geometry, Commutative Algebra and Non-commutative algebra.

This is a procedure in "Singular" for computing blowups.

blowUp2(J,C);
J,C = ideals,

Assume: $C$ = ideal containing $J$

Note: $C$ the ideal of the center of the blowup and $J$ the ideal of the variety Important differences to blowUp: - $V(J)$ itself is blown up, not the ambient space - $C$ is not assumed to be non-singular

Compute: the blowing up of $J$ in $C$, the exceptional locus and the blow-up map

Return: list, say l, of size at most size(C),

l[i] is the affine ring corresponding to the i-th chart each l[i] contains the ideals
- Jnew, ideal of the blownup J
- eD, ideal of the new exceptional divisor
- bM, ideal corresponding to the blowup map 

I want to understand the computation which this blowup2 gives for a particular example.

Let's take $J= <z^2-x^3y^2>$; $ C=<z,xy>$;

We are blowing up the variety $V(z^2-x^3y^2)$ along the center $V(z,xy)$

The output is as follows:

In the first affine chart:

basering;
==> //   characteristic : 0
==> //   number of vars : 3
==> //                  : names    x(1) x(3) y(2)

Jnew;
==> Jnew[1]=x(1)y(2)^2-1

eD;
==> eD[1]=x(3)
==> eD[2]=x(1)y(2)^2-1

bM;
==> bM[1]=x(1)
==> bM[2]=x(3)y(2)^3
==> bM[3]=x(3)

In the second affine chart:

basering;
==> //   characteristic : 0
==> //   number of vars : 2
==> //                  : names    x(2) y(1)

Jnew;
==> Jnew[1]=0

eD;
==> eD[1]=x(2)y(1)^2

bM;
==> bM[1]=y(1)^2
==> bM[2]=x(2)
==> bM[3]=x(2)y(1)^3

Can someone help me understanding this computation?