du Val singularity

In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type.

They were studied by Patrick du Val[1][2][3] and Felix Klein.

; equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.

[4] 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.

[5][6] The possible Du Val singularities are (up to analytical isomorphism):

Du Val singularies are classified by the simply laced Dynkin diagrams , a form of ADE classification .