In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...

, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its

blowup ''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemming ...

at the 10 Eckardt points.

Definition

The Clebsch surface is the set of points (''x''₀:''x''₁:''x''₂:''x''₃:''x''₄) of P⁴ satisfying the equations :

x_0 + x_1 + x_2 + x_3 + x_4 = 0,

x_0^3 + x_1^3 + x_2^3 + x_3^3 + x_4^3 = 0.

Eliminating ''x''₀ shows that it is also isomorphic to the surface :

x_1^3 + x_2^3 + x_3^3 + x_4^3 = (x_1 + x_2 + x_3 + x_4)^3

in P³.

Properties

The symmetry group of the Clebsch surface is the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...

''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. 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 points where 3 lines meet, given by the point (1 : −1 : 0 : 0 : 0) and its conjugates under permutations. showed that the surface obtained by blowing up the Clebsch surface in its 10 Eckardt points is the

Hilbert modular surface In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular varie ...

of the level 2 principal congruence subgroup of the Hilbert modular group of the field ''Q''(). 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 mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...

in 6 points. 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 In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

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.

References

* * * *

External links

*{{mathworld, urlname=ClebschDiagonalCubic, title= Clebsch diagonal cubic
Clebsch Surface

John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...

, 1 March, 2016, AMS Visual Insight Blog Algebraic surfaces