Background Information
Cubic Surfaces
A cubic surface is the vanishing set of a homogenous polynomial of degree 3 in P^{3}, i.e. it consists of all (x:y:z:w) in P^{3} with:
a_{0}x^{3} + a_{1}x^{2}y + a_{2}x^{2}z + ... + a_{18}z^{2}w + a_{19}w^{3} = 0
To give an example, here is a picture of the so called Clebsch Diagonal Surface: x^{3} + y^{3} + z^{3} + w^{3}  (x+y+z+w)^{3}. To be able to draw it, one must pass from the projective space P^{3} to the affine space. For the picture below, we set w = 2*(1xyz):


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

