Clebsch surface

In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by Clebsch (1871) and Klein (1873), all of whose 27 exceptional lines can be defined over the real numbers.

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.

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.