Cubic Surfaces
Cubic Surfaces |
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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 (1) |
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):
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 - also pictured above -, 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 have studied cubic surfaces and the structure of the 27 lines upon it.
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. For the Cubic Surface Program, we use Coble's hexahedral from, which allows us to calculate the equations of the 27 lines directly from the equation (which is not possible for a cubic surface given in the form (1) introduced at the beginning).
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, because the Galois group of such a polynomial is not solvable. Klein and Coble, however, found strategies to solve the problem by dividing it in an algebraic and a transcendental part.
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© Oliver Labs, The Cubic Surface Homepage, Algebraic Geometry Group, University of Mainz, Germany