Resolving the singularity

We can 'resolve' the singularity in ℂ²/Γ. Roughly, this means we can find a smooth 2-dimensional algebraic variety S and an onto map:

that's one-to-one away from the singularity. There are many resolutions, but one minimal resolution. All others factor uniquely through this one:

More precisely, if X is an algebraic variety with singular points X_sing ⊂ X, we say π: S → X is a resolution of X if S is smooth, π is proper, π^-1(X - X_sing) is dense in S, and π is an isomorphism between π^-1(X - X_sing) and X - X_sing. For more, see Section 6 here:

Peter Slodowy, Platonic solids, Kleinian singularities, and Lie groups, in Algebraic Geometry, Springer, Berlin, 1983, pp. 102–138.

and Section 5.2 here:

Joris van Hoboken, Platonic Solids, Binary Polyhedral Groups, Kleinian Singularities and Lie Algebras of Type A, D, and E, Master's Thesis, University of Amsterdam, 2002.

Unfortunately these sources omit the uniqueness clause in the definition of minimal resolution. For a more detailed treatment, see:

Klaus Lamotke, Regular Solids and Isolated Singularities, Vieweg & Sohn, Braunschweig, 1986.