David Madore's WebLog: The twenty-seven lines on the cubic surface

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The twenty-seven lines on the cubic surface

[Clebsch Cubic surface]I spent a good part of the afternoon creating this image (click to enlarge) of one of the remarkable inhabitants of the platonic heaven: the Clebsch cubic surface; specifically, this is the Euclidean form of the latter which has the greatest possible group of symmetries (24 of the 120 symmetries of the Clebsch cubic are realized as Euclidean isometries).

Every smooth cubic surface has twenty-seven lines on it (sometimes poetically known as Solomon's seal; I do not know who coined the term or whether it is related to the plant Polygonatum biflorum, which also goes by that name). But in general they exist only as complex lines and might not all be realized: the number of real lines can be three, seven, fifteen or twenty-seven, and on the Clebsch cubic all twenty-seven lines exist in a real sense. You can only see twenty-four lines in the picture (can you?), however, because the last three lines are away at infinity.

Furthermore, it is possible for three lines on a cubic surface to all meet in a single point, in which case the point in question is known as an Eckardt point (this is a remarkable feature, and while all cubic surfaces have lines on them, not all have Eckardt points, even in the complex sens): the Clebsch cubic surface is unique in that it has ten Eckardt points, and all are real (on my particular Euclidean realization, four are the vertices of a regular tetrahedron, two of which can be clearly seen, and six are at infinity).

Another way to represent a smooth cubic surface (at least one which has all twenty-seven lines real) is as a set of six points in the (projective) plane (in general position, that is, such that no three are aligned and all six do not line on a common conic). It is not easy to describe precisely the relation between the six marked (or blown up) points and the cubic surface[#], but it is quite easy to explain how the twenty-seven exceptional lines are seen: consider the six marked points, the fifteen lines connecting any two of them, and the six conics going through five of the marked points — now 6+15+6=27, and they correspond exactly to the exceptional lines on the cubic surface; and even intersection is preserved if we agree that intersection at the marked points in the plane is only taken into account when it is tangential[#2]. Eckardt points are also easily seen that way: when three lines defined by three pairs of marked points meet in a common point, that point is an Eckardt point; also when the conic through five of the six marked points has a tangent at one of said marked points which goes through the sixth, then that tangent direction is an Eckardt “point”. Under this correspondance, the Clebsch cubic is the most remarkable configuration of six points in the plane, namely, a regular pentagon and its center. The ten Eckardt points are then obvious.

So one of the answers I might give when asked what my thesis is about is: six points in the plane.

[#] Each point on the cubic surface corresponds to a point on the plane and, if it is one of the six marked points, a line direction throught that point. (This is what is meant by blowing up: replacing a marked point by the set of all directions through that point.)

[#2] For example, given three of the six marked points, the three lines connecting them are thought not to intersect; however, each of them intersects the two marked points which it joins. This is in accordance with the idea that the marked points have been replaced by the set of directions through them.

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