Ferromagnetic material properties

2013-11-07T20:30:03Z

74.76.214.59: /* Formulae */

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 [[ADE classification|A-D-E singularity]] type. They are the [[canonical singularities]] (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by {{harvs|txt|first=Patrick |last=du Val|authorlink=Patrick du Val|year1=1934a|year2=1934b|year3=1934c}} and [[Felix Klein]].

The du Val singularities also appear as quotients of '''C'''2 by a finite subgroup of [[SL2(C)|''SL''2('''C''')]]; equivalently, a finite subgroup of SU(2), which are known as [[binary polyhedral group]]s. The rings of [[invariant polynomial]]s 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 ==
[[File:Simply Laced Dynkin Diagrams.svg|thumb|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 analytic isomorphism):
*''A''''n'': <math>w^2+x^2+y^{n+1}=0</math>
*''D''''n'': <math> w^2+y(x^2+y^{n-2}) = 0 </math> (''n''≥4)
*''E''6: <math>w^2+x^3+y^4=0 </math>
*''E''7: <math> w^2+x(x^2+y^3)=0 </math>
*''E''8: <math> w^2+x^3+y^5=0. </math>

==See also==

*[[Brieskorn–Grothendieck resolution]]

==References==
*{{Citation | 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}}
*{{Citation | 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 | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4}}
*{{Citation | last1=Durfee | first1=Alan H. | title=Fifteen characterizations of rational double points and simple critical points | url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1979:25::300 | mr=543555 | year=1979 | journal=L'Enseignement Mathématique. Revue Internationale. IIe Série | issn=0013-8584 | volume=25 | issue=1 | pages=131–163}}
*{{citation|first= Patrick|last= du Val|title=On isolated singularities of surfaces which do not affect the conditions of adjunction. I|journal= Proceedings of the Cambridge Philosophical Society|volume= 30 |year=1934a|pages= 453–459, |doi=10.1017/S030500410001269X|issue= 4}} [http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17602 Zbl entry I]
*{{citation|first= Patrick|last= du Val|title=On isolated singularities of surfaces which do not affect the conditions of adjunction. II|journal= Proceedings of the Cambridge Philosophical Society|volume= 30 |year=1934b|pages= 460–465|doi=10.1017/S0305004100012706|issue= 4}} [http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17603 II]
*{{citation|first= Patrick|last= du Val|title=On isolated singularities of surfaces which do not affect the conditions of adjunction. III|journal= Proceedings of the Cambridge Philosophical Society|volume= 30 |year=1934c|pages= 483–491|doi=10.1017/S030500410001272X|issue= 4}} [http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0010.17701 III]

==External links==
*{{citation|authorlink=M. Reid|first=M. |last=Reid|url=http://www.warwick.ac.uk/~masda/surf/more/DuVal.pdf |title=The du Val singularities ''A''''n'', ''D''''n'', ''E''6, ''E''7, ''E''8}}
*{{citation|title=Du Val Singularities|first= Igor|last= Burban|url=http://www.mi.uni-koeln.de/~burban/singul.pdf }}

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[[Category:Algebraic surfaces]]
[[Category:Singularity theory]]

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Ferromagnetic material properties