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 are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by Patrick du Val^[1][2][3] and Felix Klein.

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.^[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]