Background Information

A Brief History

The 27 Lines on Cubic Surfaces

The Double Six Configuration

The Singularities

Literature and Links
Rational Double Points Other Singularities Tools Bibliography                  

Background Information

Cubic Surfaces

A cubic surface is the vanishing set of a homogenous polynomial of degree 3 in P3, i.e. it consists of all (x:y:z:w) in P3 with:

a0x3 + a1x2y + a2x2z + ... + a18z2w + a19w3 = 0

To give an example, here is a picture of the so called Clebsch Diagonal Surface: x3 + y3 + z3 + w3 - (x+y+z+w)3. To be able to draw it, one must pass from the projective space P3 to the affine space. For the picture below, we set w = 2*(1-x-y-z):

The Clebsch Diagonal Cubic Surface.
It is smooth and all its different 27 lines are real.
There are 10 so-called Eckardt points, i.e. points in which three of the lines meet.