2001:638:208:FD5F:FCF4:C8CA:57D7:B740: /* Uniqueness */

2014-01-18T16:02:32Z

Uniqueness

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{{no footnotes|date=August 2012}}
[[File:Clebsch Cublic.png|thumb|The Clebsch Cubic in a local chart]]
[[File:Modell der Diagonalfläche von Clebsch -Schilling VII, 1 - 44-.jpg|thumb|right|Model of the surface]]
In mathematics, the '''Clebsch diagonal cubic surface''', or '''Klein's icosahedral cubic surface''' is a [[cubic surface]] studied by {{harvtxt|Clebsch|1871}} and {{harvtxt|Klein|1873}} all of whose 27 [[exceptional line]]s
can be defined over the real numbers. The term '''Klein's icosahedral surface''' can refer to either this surface or its blowup at the 10 [[Eckardt point]]s.

==Definition==

The Clebsch surface is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4) of ''P''4 satisfying the equations
:<math>\displaystyle x_0+x_1+x_2+x_3+x_4 = 0</math>
:<math>\displaystyle x_0^3+x_1^3+x_2^3+x_3^3+x_4^3 = 0.</math>

Eliminating ''x''0 shows that it is also isomorphic to the surface
:<math>\displaystyle x_1^3+x_2^3+x_3^3+x_4^3 = (x_1+x_2+x_3+x_4)^3</math>
in ''P''3.

The symmetry group of the surface is the [[symmetric group]] ''S''5 of order 120, acting by permutations of the coordinates (in ''P''4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group.

==Properties==

The 27 exceptional lines are:
* The 15 images (under ''S''5) of the line of points of the form (''a'' : −''a'' : ''b'' : −''b'' : 0).
* The 12 images of the line though the point (1:ζ: ζ2: ζ3: ζ4) and its complex conjugate, where ζ is a primitive 5th root of 1.

The surface has 10 [[Eckardt point]]s where 3 lines meet, given by the point
(1 : −1 : 0 : 0 : 0) and its conjugates under permutations. {{harvtxt|Hirzebruch|1976}} showed that the surface obtained by blowing up the Clebsch surface in its 10 Eckardt points is the [[Hilbert modular surface]] of the level 2 principal congruence subgroup of the Hilbert modular group of the field ''Q''(√5). The quotient of the Hilbert modular group by its level 2 congruence subgroup is isomorphic to the alternating group of order 60 on 5 points.

Like all nonsingular cubic surfaces, the Clebsch cubic can be obtained by blowing up the projective plane in 6 points. {{harvtxt|Klein|1873}} described these points as follows. If the projective plane is identified with the set of lines through the origin in a 3-dimensional vector space containing an icosahedron centered at the origin, then the 6 points correspond to the 6 lines through the centers of the 12 vertices. The Eckardt points correspond to the 10 lines through the centers of the 20 faces.

==References==
*{{Citation | last1=Clebsch | first1=A. | authorlink = Alfred Clebsch | title= Ueber die Anwendung der quadratischen Substitution auf die Gleichungen 5ten Grades und die geometrische Theorie des ebenen Fünfseits | doi=10.1007/BF01442599 | year=1871 | journal=Mathematische Annalen | volume=4 | issue=2 | pages=284–345}}
*{{Citation | last1=Hirzebruch | first1=Friedrich | author1-link=Friedrich Hirzebruch | title=The Hilbert modular group for the field Q(√5), and the cubic diagonal surface of Clebsch and Klein | doi=10.1070/RM1976v031n05ABEH004190 | mr=0498397 | year=1976 | journal=Russian Math. Surveys | issn=0042-1316 | volume=31 | issue=5 | pages=96–110}}
*{{Citation | last1=Hunt | first1=Bruce | title=The geometry of some special arithmetic quotients | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-61795-2 | doi=10.1007/BFb0094399 | mr=1438547 | year=1996 | volume=1637}}
*{{Citation | last1=Klein | first1=Felix | title=Ueber Flächen dritter Ordnung | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01443196 | year=1873 | journal=Mathematische Annalen | volume=6 | issue=4 | pages=551}}

==External links==
*{{citation |first=Robert |last=Ferréol |first2= L. G. |last2=Vidiani |first3= Alain |last3= Esculier |year=2004 |url=http://www.mathcurve.com/surfaces/clebsch/clebsch.shtml |title=Surface de Clebsch}}
*{{mathworld|urlname=ClebschDiagonalCubic|title= Clebsch diagonal cubic}}

[[Category:Algebraic surfaces]]

Doob decomposition theorem - Revision history

2001:638:208:FD5F:FCF4:C8CA:57D7:B740: /* Uniqueness */