Du Val singularity

The Du Val singularities also appear as quotients of ${\displaystyle {\mathbb{C}}^2}$ by a finite subgroup of SL₂${\displaystyle (\mathbb{C})}$; equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups. The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory.

Classification

The possible Du Val singularities are (up to analytical isomorphism):

• ${\displaystyle A_n:w^2+x^2+y^{n+1}=0}$
• ${\displaystyle D_n:w^2+y(x^2+y^{n-2})=0(n\ge 4)}$
• ${\displaystyle E_6:w^2+x^3+y^4=0}$
• ${\displaystyle E_7:w^2+x(x^2+y^3)=0}$
• ${\displaystyle E_8:w^2+x^3+y^5=0.}$