Clebsch surface

 ( Algebraic Surfaces ) 

In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowing up at the 10 .

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Definition The Clebsch surface is the set of points ( x₀: x₁: x₂: x₃: x₄) of projective space 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 x₀ 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. In , it can be represented by

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Properties The symmetry group of the Clebsch surface is the symmetric group S₅ of order 120, acting by permutations of the coordinates (in P⁴). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group.

The 27 exceptional lines are:

• The 15 images (under S₅) 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 unity.

The surface has 10 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 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 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 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.

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Explicit description of lines Using the embedding (), the 27 lines are given by , where , , , , , and are all taken from the same row in the following table:

+ ! !! !! !! !! !!
$-\backslash frac\{1\}\{3\}$
$0$
$0$
$0$
$\backslash frac\{1\}\{3\}$
$\backslash frac\{1\}\{6\}$
$\backslash frac\{1\}\{3\}$
$\backslash frac\{1\}\{6\}$
$\backslash frac\{1\}\{3\}$
$\backslash frac\{1\}\{6\}$
$0$
$\backslash frac\{1\}\{6\}$
$0$
$\backslash frac\{1\}\{6\}$
$\backslash frac\{1\}\{6\}$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$
$0$

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Further reading

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External links

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