ADE singularity

Remarks

Equivalently, the germ is isomorphic to the quotient $\C^2/G$ at the image of $0$, for a nontrivial finite subgroup $G$ of the special linear group $SL(2, \C)$. The group is cyclic of order $n + 1$ for $A_n$, and binary dihedral of order $4(n - 2)$ for $D_n$. For $E_6$, $E_7$, $E_8$ it is binary tetrahedral, octahedral and icosahedral, of orders $24$, $48$ and $120$. The exceptional curves of the minimal resolution are smooth rational curves with self-intersection $-2$, and their dual graph is the Dynkin diagram of the same name. ADE singularities are rational singularities, and they are exactly the rational double points. Example: $G = \set{\pm 1}$ acts on $\C[u, v]$ with invariant ring generated by $a = u^2$, $b = v^2$, $c = uv$, subject to $ab = c^2$. After the linear change $a = x + iy$, $b = -(x - iy)$, $c = z$, the relation $ab - c^2 = 0$ becomes $x^2 + y^2 + z^2 = 0$, which is the $A_1$ singularity. Non-example: the cone $x^3 + y^3 + z^3 = 0$ has a singular point at $0$ of multiplicity $3$, while every ADE singularity has multiplicity $2$, so it is not an ADE singularity.