(ADE singularities)
An ADE singularity, also called a du Val singularity or rational double point, is a surface singularity over \(\CC\) analytically isomorphic to the singularity at the origin of one of the following hypersurfaces in \(\AA^3\):
| Type | Equation | Group \(G \subseteq \SL_2(\CC)\) |
|---|---|---|
| \(A_n\), \(n \geq 1\) | \(x^2 + y^2 + z^{n+1} = 0\) | cyclic of order \(n+1\) |
| \(D_n\), \(n \geq 4\) | \(x^2 + y^2 z + z^{n-1} = 0\) | binary dihedral of order \(4(n-2)\) |
| \(E_6\) | \(x^2 + y^3 + z^4 = 0\) | binary tetrahedral of order \(24\) |
| \(E_7\) | \(x^2 + y^3 + y z^3 = 0\) | binary octahedral of order \(48\) |
| \(E_8\) | \(x^2 + y^3 + z^5 = 0\) | binary icosahedral of order \(120\) |
The ADE singularities are exactly the quotient singularities \(\CC^2 / G\) for finite subgroups \(G \subseteq \SL_2(\CC)\), with \(G\) as in the table.
They are exactly the rational double points, and exactly the canonical surface singularities.
The exceptional locus of the minimal resolution is a union of smooth rational curves of self-intersection \(-2\) whose dual graph is the Dynkin diagram of the same name, and the minimal resolution is crepant.
\(A_1\) is the quadric cone \(x^2 + y^2 + z^2 = 0\), the quotient \(\CC^2 / \{\pm 1\}\); its minimal resolution is the blowup of the vertex, with one \((-2)\)-curve.
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