Du Val singularity

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[1][2][3] and Felix Klein.

The Du Val singularities also appear as quotients of C 2 {\displaystyle \mathbb {C} ^{2}} by a finite subgroup of SL2 ( C ) {\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):

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

See also

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. 25 (1). European Mathematical Society Publishing House: 131–163. doi:10.5169/seals-50375. ISSN 0013-8584. MR 0543555. Archived from the original on 2022-05-09. Retrieved 2022-05-09.

External links

  • Reid, Miles, The Du Val singularities An, Dn, E6, E7, E8 (PDF)
  • Burban, Igor, Du Val Singularities (PDF)