I'v been trying to understand how Du Val singularities are resolved and what the exceptional divisors look like so I can work out their dynkin diagrams. A basic example I tried in the interest of getting some intuition was $z^2+y(x+y^2)=0$. First blow it up with coordinates $[x_0:y_0:z_0]$ and let $x_0=1$ so $y=xy_0$ , $z=xz_0$ to so we should get one of the affine charts $(xz_0)^2+xy_0(x+(xy_0)^2)=0$,
$\implies x^2(z_0^2+y_0+xy_0^3)=0$
My understanding is that the the exceptional divisor is set of points which are mapped to by $(0,0,0)$ during the blow up, so were looking for solutions of $x^2(z_0^2+y_0+xy_0^3)=0$ where $x=y=z=0$, but wouldn't that just be the plane $(0,y_0,z_0)$? At this point I'm not sure how to proceed as I don't know if this is actually the correct exceptional divisor.
If anyone could give a general explication of how to find exceptional divisors or show how to fully resolve this example and find all exceptional divisors and how they intersect that would be great. Thanks.