---
tag: HQNXQBZ6C7
kind: concept
name: ADE singularity
msc: [14J17, 32S25, 14B05]
specializes: [98QASBAMEB]
sources:
  - url: https://en.wikipedia.org/wiki/Du_Val_singularity
same_as:
  wikidata: Q5310086
  wikipedia: Du Val singularity
---

## Summary

The ADE singularities, also called Du Val singularities, simple surface singularities or rational double points, are the simplest singular points of complex surfaces.
They are the quotients $\C^2/G$ by finite subgroups $G$ of the [[special linear group|0V9PMN0RBH]] $SL(2, \C)$.
They come in two infinite families $A_n$ and $D_n$ and three exceptional cases $E_6$, $E_7$, $E_8$, matching the simply laced Dynkin diagrams.
The match is visible in the minimal resolution, whose exceptional curves meet in the pattern of the Dynkin diagram.

## Explanations

### L1 human

The simplest example is the cone $x^2 + y^2 + z^2 = 0$, the $A_1$ singularity, which is the quotient of the plane by $v \mapsto -v$.
Blowing up the tip of the cone replaces it by one smooth rational curve.
In general the $A_n$ singularity is resolved by a chain of $n$ such curves, each meeting the next one, just like the Dynkin diagram $A_n$.
The same ADE list classifies finite rotation groups in three dimensions and simple Lie algebras of simply laced type.
This is why these singularities are a crossroads of mathematics.