The Clebsch Map

Now comes a very interesting observation. There is a new kind of mapping that arises from a trihedral pair.

The mappings that we will study are called birational maps. They can be expressed using fractions of polynomials (rational expressions), hence the name.

A chosen trihedral pair allows to write the equation of the surface as a sum of two products of three linear terms each (in 4 variables).

This expression can be rewritten in 72 ways as a determinantal identity involving a three by three matrix whose entries are the linear terms (or zero). There are 72 matrices which can be formed such that the vanishing of the determinant is the same as the equation of the surface being zero.

Pick any of the 72 matrices.

We can now write the condition that a point lies on the surface as a vanishing of the determinant of the matrix, which in turn is equivalent to the fact that the rows (or columns) of the matrix are linearly dependent.

Thus, up to scalar multiples, a point on the surface can be mapped to a point of the projective plane (three coefficients determine a point in the projective plane).

This mapping has the following properties:

There are exactly six lines on the surface which are "contracted". These six lines are pairwise disjoint. Each of these lines maps to one point (in the sense that all points on the line map to the same point in the plane). Outside the six lines and outside the six points in the plane, the map is one-to-one.

File translated from T_EX by T_TH, version 4.08.
On 4 Jun 2017, 11:23.