Resolving the singularity previous next

We can 'resolve' the singularity in 2/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: and Section 5.2 here: Unfortunately these sources omit the uniqueness clause in the definition of minimal resolution. For a more detailed treatment, see: