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 Xsing ⊂ X, we say π: S → X is a resolution of X if S is smooth, π is proper, π-1(X - Xsing) is dense in S, and π is an isomorphism between π-1(X - Xsing) and
X - Xsing. For more, see Section 6 here:
Peter Slodowy, Platonic solids, Kleinian singularities, and Lie groups, in Algebraic Geometry, Springer, Berlin, 1983, pp. 102–138.