A Brief History
There are many more important developments in the history of cubic surfaces than those touched below. We hope to include more information later... but we will be very happy if you contact us in order to contribute to this section!
The Important First Steps
In the 19th century, mathematicians started to study the structure of such vanishing sets of polynomials of different degrees in P3, called algebraic surfaces. It turned out, that each generic cubic surface contains 27 straight lines, which are cut out of the surface in sets of three by 45 so-called tritangent planes.
From this starting point, a lot of mathematicians studied cubic surfaces and the structure of the 27 lines upon it.
The Problem of Computing the Lines
In 1861, Clebsch showed, that the defining equation of a cubic surface can be put, in a unique way, in the so called pentahedral form (i.e. a sum of fix cubes of linear polynomials). For the Cubic Surface Program xcsprg, we use Coble's hexahedral from (i.e. sum of six cubes of linear polynomials), which allows us to calculate the equations of the 27 lines directly from the equation.
In order to know the equations of the lines, one would like to be able to pass from one form of the equation to the other, but there is no algebraic way to pass from the form (1) to one of the others. Indeed, one can reduce the problem to a polynomial of degree 27, which can not be solved purely algebraicly,
In the 20th century, it became possible to understand the beauty of cubic surfaces in a much better way.
E.g., the combinations of singularities which can occur on a cubic surface can simply be described as all possible sub-graphs of the Coxeter diagram of the E6~ singularity.
... we hope to work this out in detail in the near future...