du Val singularity

{{short description|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{{cite journal|first=Patrick|last=du Val|authorlink=Patrick du Val| title=On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry I|journal=Proceedings of the Cambridge Philosophical Society|volume=30|year=1934a|pages=453–459|doi=10.1017/S030500410001269X|issue=4|s2cid=251095858 |url=https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17602|archive-url=https://web.archive.org/web/20220509161614/https://zbmath.org/?q=an%3A0010.17602|archive-date=9 May 2022|url-access=subscription}}{{cite journal|first=Patrick|last= du Val|authorlink=Patrick du Val|title=On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry II|journal=Proceedings of the Cambridge Philosophical Society|volume=30|year=1934b|pages=460–465|doi=10.1017/S0305004100012706|issue=4|s2cid= 197459819|url=https://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17603|archive-url=https://web.archive.org/web/20220513091059/https://zbmath.org/?q=an%3A0010.17603|archive-date=13 May 2022|url-access=subscription}}{{cite journal|first=Patrick|last=du Val|authorlink=Patrick du Val|title=On isolated singularities of surfaces which do not affect the conditions of adjunction, Entry III|journal=Proceedings of the Cambridge Philosophical Society|volume=30|year=1934c|pages=483–491|doi=10.1017/S030500410001272X|issue=4|s2cid=251095521 |url=http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17701|archive-date=9 May 2022|archive-url=https://web.archive.org/web/20220509164515/https://zbmath.org/?q=an%3A0010.17701|url-access=subscription}} and Felix Klein.

The Du Val singularities also appear as quotients of $\backslash Complex^2$ by a finite subgroup of SL₂$(\backslash Complex)$; equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups.{{cite book|last1=Barth|first1=Wolf P.|last2=Hulek|first2=Klaus|last3=Peters|first3=Chris A.M.|last4=Van de Ven|first4=Antonius|title=Compact Complex Surfaces|pages=197–200|publisher=Springer-Verlag, Berlin|series=Ergebnisse der Mathematik und ihre Grenzbereiche. 3. Teil (Results of mathematics and their border areas. 3rd Part)|url=https://books.google.com/books?id=LtWDVZxiK6EC|oclc=642357691|isbn=978-3-540-00832-3|mr=2030225|year=2004|volume=4|access-date=2022-05-09|archive-date=2022-05-09|archive-url=https://web.archive.org/web/20220509164511/https://www.google.co.in/books/edition/Compact_Complex_Surfaces/LtWDVZxiK6EC?hl=en|url-status=live}} 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.{{cite journal|last1=Artin |first1=Michael|author1-link=Michael Artin|title=On isolated rational singularities of surfaces| jstor=2373050|mr=0199191| year=1966|journal=American Journal of Mathematics| issn=0002-9327|volume=88|pages=129–136| doi=10.2307/2373050|issue=1}}{{Cite journal |last1=Durfee |first1=Alan H. |year=1979 |title=Fifteen characterizations of rational double points and simple critical points |url=https://www.e-periodica.ch/digbib/view?pid=ens-001:1979:25::59#300 |journal=L'Enseignement mathématique |series=IIe Série |publisher=European Mathematical Society Publishing House |volume=25 |issue=1 |pages=131–163 |doi=10.5169/seals-50375 |issn=0013-8584 |mr=543555 |access-date=2022-05-09 |archive-date=2022-05-09 |archive-url=https://web.archive.org/web/20220509163159/https://www.e-periodica.ch/digbib/view?pid=ens-001:1979:25::59#300 |url-status=live }}