Resolution of ADE- singularities: The second blow-up for the A_2 singularity

Question

I try to resolve the surface singularity $X\subset \mathbb{A}^3$ of type $A_2$. It is defined by $x^2+y^2+z^3=0$. I expect to need at least two blow-ups. But after the first blow-up of $X$ the strict transform $\tilde{X}$ seems to be already non-singular: As usual I check $$\tilde{X} \cap (\mathbb{A}^3\times \mathbb{P}^2)$$ with respect to each of three affine standard sets of $\mathbb{P}^2$.

What is wrong? Do I have to blow-up for the second blow-up an infinitesimal near point, i.e. a point on the exceptional divisor of the first blow-up?

Answer 1

Indeed: The second blow-up has to blow-up a singular point of multiplicity $=2$ of the exceptional divisor $E_1$ of the first blow-up. Hence one gets a second exceptional divisor $E_2$ with self-intersection -2, and both exceptional divisors have intersection number $=1$.The dual graph of the two exceptional divisors is the Coxeter graph $A_2$ with annotation -2.