# 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):

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

## References

1. ^ du Val, Patrick (1934a). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry I". Proceedings of the Cambridge Philosophical Society. 30 (4): 453–459. doi:10.1017/S030500410001269X. S2CID 251095858. Archived from the original on 9 May 2022.
2. ^ du Val, Patrick (1934b). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry II". Proceedings of the Cambridge Philosophical Society. 30 (4): 460–465. doi:10.1017/S0305004100012706. S2CID 197459819. Archived from the original on 13 May 2022.
3. ^ du Val, Patrick (1934c). "On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry III". Proceedings of the Cambridge Philosophical Society. 30 (4): 483–491. doi:10.1017/S030500410001272X. S2CID 251095521. Archived from the original on 9 May 2022.
4. ^ Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004). Compact Complex Surfaces. Ergebnisse der Mathematik und ihre Grenzbereiche. 3. Teil (Results of mathematics and their border areas. 3rd Part). Vol. 4. Springer-Verlag, Berlin. pp. 197–200. ISBN 978-3-540-00832-3. MR 2030225. OCLC 642357691. Archived from the original on 2022-05-09. Retrieved 2022-05-09.
5. ^ Artin, Michael (1966). "On isolated rational singularities of surfaces". American Journal of Mathematics. 88 (1): 129–136. doi:10.2307/2373050. ISSN 0002-9327. JSTOR 2373050. MR 0199191.
6. ^ Durfee, Alan H. (1979). "Fifteen characterizations of rational double points and simple critical points". L'Enseignement mathématique. IIe Série. European Mathematical Society Publishing House. 25 (1): 131–163. doi:10.5169/seals-50375. ISSN 0013-8584. MR 0543555. Archived from the original on 2022-05-09. Retrieved 2022-05-09.