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]

Classification

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

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

• ${\displaystyle A_{n}:\quad w^{2}+x^{2}+y^{n+1}=0}$
• ${\displaystyle D_{n}:\quad w^{2}+y(x^{2}+y^{n-2})=0\qquad (n\geq 4)}$
• ${\displaystyle E_{6}:\quad w^{2}+x^{3}+y^{4}=0}$
• ${\displaystyle E_{7}:\quad w^{2}+x(x^{2}+y^{3})=0}$
• ${\displaystyle E_{8}:\quad w^{2}+x^{3}+y^{5}=0.}$

See also

• Brieskorn–Grothendieck resolution
• General elephant conjecture