98:
25:
90:
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centered at the origin, then the 6 points correspond to the 6 lines through the icosahedron's 12 vertices. The
Eckardt points correspond to the 10 lines through the centers of the 20 faces.
54:
656:
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
408:
97:
673:(1871), "Ueber die Anwendung der quadratischen Substitution auf die Gleichungen 5ten Grades und die geometrische Theorie des ebenen Fünfseits",
645:). 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.
753:
285:
76:
626:
where 3 lines meet, given by the point (1 : −1 : 0 : 0 : 0) and its conjugates under permutations.
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The 12 images of the line though the point (1:ζ: ζ: ζ: ζ) and its complex conjugate, where ζ is a primitive 5th root of 1.
37:
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696:(1976), "The Hilbert modular group for the field Q(√5), and the cubic diagonal surface of Clebsch and Klein",
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showed that the surface obtained by blowing up the
Clebsch surface in its 10 Eckardt points is the
819:
585:). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group.
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Like all nonsingular cubic surfaces, the
Clebsch cubic can be obtained by blowing up the
634:
of the level 2 principal congruence subgroup of the
Hilbert modular group of the field
670:
833:
623:
549:{\displaystyle x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=(x_{1}+x_{2}+x_{3}+x_{4})^{3}}
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385:{\displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0.}
736:, Lecture Notes in Mathematics, vol. 1637, Berlin, New York:
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of order 120, acting by permutations of the coordinates (in
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734:The geometry of some special arithmetic quotients
571:The symmetry group of the Clebsch surface is the
402:shows that it is also isomorphic to the surface
274:{\displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0,}
46:but its sources remain unclear because it lacks
129:can be defined over the real numbers. The term
8:
782:(4), Springer Berlin / Heidelberg: 551–581,
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149:The Clebsch surface is the set of points (
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77:Learn how and when to remove this message
16:Non-singular cubic surface in mathematics
826:, 1 March, 2016, AMS Visual Insight Blog
133:can refer to either this surface or its
118:
599:) of the line of points of the form (
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7:
93:The Clebsch cubic in a local chart
14:
111:Klein's icosahedral cubic surface
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772:"Ueber Flächen dritter Ordnung"
710:10.1070/RM1976v031n05ABEH004190
588:The 27 exceptional lines are:
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107:Clebsch diagonal cubic surface
1:
131:Klein's icosahedral surface
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190:satisfying the equations
806:"Clebsch diagonal cubic"
32:This article includes a
632:Hilbert modular surface
61:more precise citations.
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776:Mathematische Annalen
770:Klein, Felix (1873),
698:Russian Math. Surveys
694:Hirzebruch, Friedrich
675:Mathematische Annalen
592:The 15 images (under
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276:
100:
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732:Hunt, Bruce (1996),
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113:, is a non-singular
105:In mathematics, the
101:Model of the surface
622:The surface has 10
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840:Algebraic surfaces
803:Weisstein, Eric W.
788:10.1007/BF01443196
746:10.1007/BFb0094399
687:10.1007/BF01442599
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125:, all of whose 27
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34:list of references
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628:Hirzebruch (1976)
127:exceptional lines
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38:related reading
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681:(2): 284–345,
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652:in 6 points.
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117:, studied by
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115:cubic surface
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654:Klein (1873)
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396:
395:Eliminating
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123:Klein (1873)
110:
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73:
64:
53:Please help
45:
671:Clebsch, A.
658:icosahedron
615: : 0).
67:August 2012
59:introducing
664:References
567:Properties
145:Definition
137:at the 10
824:John Baez
811:MathWorld
718:0042-1316
834:Category
607: :
764:1438547
726:0498397
640:√
55:improve
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752:
724:
716:
135:blowup
184:) of
109:, or
40:, or
750:ISBN
714:ISSN
121:and
784:doi
742:doi
706:doi
683:doi
559:in
836::
822:,
808:.
778:,
774:,
760:MR
758:,
748:,
740:,
722:MR
720:,
712:,
702:31
700:,
677:,
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780:6
744::
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594:S
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579:5
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51:.
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