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  "body": "\n## Definition\n\nLet $X$ be a complex surface and let $p \\in X$ be a [[singular point|D86X15A8WS]].\nThe point $p$ is an *ADE singularity*, or *Du Val singularity*, if the germ $(X, p)$ is analytically isomorphic to a germ $(\\set{f = 0}, 0)$ in $\\C^3$.\nHere $f$ is one of the following polynomials:\n\n1. $A_n$, for $n \\ge 1$: $f = x^2 + y^2 + z^{n+1}$.\n2. $D_n$, for $n \\ge 4$: $f = x^2 + y^2 z + z^{n-1}$.\n3. $E_6$: $f = x^2 + y^3 + z^4$.\n4. $E_7$: $f = x^2 + y^3 + y z^3$.\n5. $E_8$: $f = x^2 + y^3 + z^5$.\n\n## Remarks\n\nEquivalently, 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|0V9PMN0RBH]] $SL(2, \\C)$.\nThe group is cyclic of order $n + 1$ for $A_n$, and binary dihedral of order $4(n - 2)$ for $D_n$.\nFor $E_6$, $E_7$, $E_8$ it is binary tetrahedral, octahedral and icosahedral, of orders $24$, $48$ and $120$.\nThe 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.\nADE singularities are [[rational singularities|98QASBAMEB]], and they are exactly the rational double points.\nExample: $G = \\set{\\pm 1}$ acts on $\\C[u, v]$ with invariant ring generated by $a = u^2$, $b = v^2$, $c = uv$, subject to $ab = c^2$.\nAfter 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.\nNon-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.\n",
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    "name": "ADE singularity",
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