Du Val singularity

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title: "Du Val singularity" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-surfaces", "singularity-theory"] description: "Mathematical concept describing isolated singularity of an algebraic surface" topic_path: "general/algebraic-surfaces" source: "https://en.wikipedia.org/wiki/Du_Val_singularity" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept describing isolated singularity of an algebraic surface ::

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 and Felix Klein.

The Du Val singularities also appear as quotients of \Complex^2 by a finite subgroup of SL2(\Complex); 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

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/00/Simply_Laced_Dynkin_Diagrams.svg" caption="Du Val singularies are classified by the [[simply laced Dynkin diagram]]s, a form of [[ADE classification]]."] ::

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

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

References

References

1. du Val, Patrick. (1934a). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry I". [[Mathematical Proceedings of the Cambridge Philosophical Society.
2. du Val, Patrick. (1934b). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry II". [[Mathematical Proceedings of the Cambridge Philosophical Society.
3. du Val, Patrick. (1934c). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry III". [[Mathematical Proceedings of the Cambridge Philosophical Society.
4. (2004). "Compact Complex Surfaces". Springer-Verlag, Berlin.
5. (1966). "On isolated rational singularities of surfaces". [[American Journal of Mathematics]].
6. (1979). "Fifteen characterizations of rational double points and simple critical points". [[European Mathematical Society Publishing House]].

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