How to prove the minimal resolution of double rational points can be achieved by iterated blow-ups In the Slodowy's survey on Kleinian singularities, there is a statement that the minimal resolution of Kleinian singularities can be obtained by iterated blow-ups, I want to know the detail of the proof of this claim.
And I also wonder that the how to blow up the variety we obtain after the first blow-up, since it is a subvariety of $\mathbb{A}^{n}\times\mathbb{P}^{n-1}$,  should we treat it as a subvariety of $\mathbb{A}^{2n}$?
This is the link of the survey mentioned above :
http://perso.ens-lyon.fr/sandra.rozensztajn/enseignement/Slodowy.pdf
 A: First let's answer the second question : it doesn't make sense to embedd $\Bbb P^{n-1}$ in $\Bbb A^n$ (any morphism from a projective variety to an affine variety is constant). Over complex numbers, we can take a local chart and compute the blow-up in the local chart, it means that we can always suppose that the blow-up is in the affine space.
There is an elementary proof if your definition of Kleinian singularities is  quotient $X = \Bbb C^2/ \Gamma$ where $\Gamma \subset \rm{SL}_2(\Bbb C)$ is a finite subgroup. Then, you just need to compute the equations of $X$ and compute explicitely the blow-up. Computing the blow-up is not hard, and computing the equations might be tedious especially for type $E$, but it's very easy for type $A$ and not too hard in type $D$. 
If you use alternating definitions, the paper "Fifteen characterization of rational double points" by Durfee explains many characterizations of the Kleinian singularities, and prove their equivalence, in particular they are the only surfaces with a double point which can be resolved only using blow-up. But I am not sure he proves everything.
