Eckardt Points

If three lines of a surface are concurrent, the point of intersection is called an Eckardt point.

The only way that three lines of a surface can be concurrent is if they lie in a tritangent plane.

Since there are 45 tritangent planes, there are at most 45 Eckardt points.

The Clebsch surface shown above has 10 Eckardt points. Seven of them can be seen in the affine picture.

Three more lie at infinity.

Recall that we said that the three tritangent planes intersect in a line at infinity?

Well, the lines inside these tritangent planes fall into groups of three parallel lines.

Parallel lines intersect in a point at inifinity.

Since there are three sets of parallel lines, this yields three more Eckardt points.

The number of Eckardt points can be used to distinguish cubic surfaces up to isomorphism.

If two surfaces have different numbers of Eckardt points, they cannot be isomorphic. The converse is false. There are cubic surfaces with the same number of Eckardt points which are not isomorphic.

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