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