---
tag: FP32JHA9M8
kind: definition
name: ADE singularity
msc: [14J17, 32S25]
defines: HQNXQBZ6C7
uses: [D86X15A8WS, 98QASBAMEB, 0V9PMN0RBH]
sources:
  - url: https://en.wikipedia.org/wiki/Du_Val_singularity
---

## Definition

Let $X$ be a complex surface and let $p \in X$ be a [[singular point|D86X15A8WS]].
The 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 $\C^3$.
Here $f$ is one of the following polynomials:

1. $A_n$, for $n \ge 1$: $f = x^2 + y^2 + z^{n+1}$.
2. $D_n$, for $n \ge 4$: $f = x^2 + y^2 z + z^{n-1}$.
3. $E_6$: $f = x^2 + y^3 + z^4$.
4. $E_7$: $f = x^2 + y^3 + y z^3$.
5. $E_8$: $f = x^2 + y^3 + z^5$.

## 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|0V9PMN0RBH]] $SL(2, \C)$.
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|98QASBAMEB]], 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.