Du Val singularity explained

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

\Complex ^{2} by a finite subgroup of SL₂(\Complex) ; 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

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

A _{n:} w ^{2+x} ^{2+y} ^{n+1} =0 D _{n:} w ^{2+y(x} ^{2+y} ^{n-2} ) = 0 (n\ge 4 ) E _{6:} w ^{2+x} ^{3+y} ^{4=0} E _{7:} w ^{2+x(x} ^{2+y} ^{3)=0} E _{8:} w ^{2+x} ^{3+y} ^{5=0.}

See also

• Brieskorn–Grothendieck resolution
• General elephant conjecture

Notes and References

1. Patrick. du Val. Patrick du Val. On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry I. Proceedings of the Cambridge Philosophical Society. 30. 1934a. 453–459. 10.1017/S030500410001269X. 4. 251095858 . https://web.archive.org/web/20220509161614/https://zbmath.org/?q=an%3A0010.17602. 9 May 2022.
2. Patrick. du Val. Patrick du Val. On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry II. Proceedings of the Cambridge Philosophical Society. 30. 1934b. 460–465. 10.1017/S0305004100012706. 4. 197459819. https://web.archive.org/web/20220513091059/https://zbmath.org/?q=an%3A0010.17603. 13 May 2022.
3. Patrick. du Val. Patrick du Val. On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry III. Proceedings of the Cambridge Philosophical Society. 30. 1934c. 483–491. 10.1017/S030500410001272X. 4. 251095521 . 9 May 2022. https://web.archive.org/web/20220509164515/https://zbmath.org/?q=an%3A0010.17701.
4. Book: Barth. Wolf P.. Hulek. Klaus. Peters. Chris A.M.. Van de Ven. Antonius. Compact Complex Surfaces. 197–200. Springer-Verlag, Berlin. Ergebnisse der Mathematik und ihre Grenzbereiche. 3. Teil (Results of mathematics and their border areas. 3rd Part). 642357691. 978-3-540-00832-3. 2030225. 2004. 4. 2022-05-09. 2022-05-09. https://web.archive.org/web/20220509164511/https://www.google.co.in/books/edition/Compact_Complex_Surfaces/LtWDVZxiK6EC?hl=en. live.
5. Artin . Michael. Michael Artin. On isolated rational singularities of surfaces. 2373050. 0199191. 1966. American Journal of Mathematics. 0002-9327. 88. 129–136. 10.2307/2373050. 1.
6. Durfee . Alan H. . 1979 . Fifteen characterizations of rational double points and simple critical points . . IIe Série . . 25 . 1 . 131–163 . 10.5169/seals-50375 . 0013-8584 . 543555 . 2022-05-09 . 2022-05-09 . https://web.archive.org/web/20220509163159/https://www.e-periodica.ch/digbib/view?pid=ens-001:1979:25::59#300 . live .