27 Lines On A Cubic Surface
The Cayley-Salmon theorem states that a smooth cubic surface over an algebraically closed field contains 27 straight lines. These can be characterized independently of the embedding into projective space as the rational lines with self-intersection number −1, or in other words the −1-curves on the surface. An Eckardt point is a point where 3 of the 27 lines meet.
A smooth cubic surface can also be described as a rational surface obtained by blowing up six points in the projective plane in general position (in this case, “general position” means no three points are aligned and no six are on a conic section). The 27 lines are the exceptional divisors above the 6 blown up points, the proper transforms of the 15 lines in which join two of the blown up points, and the proper transforms of the 6 conics in which contain all but one of the blown up points.
Clebsch gave a model of a cubic surface, called the Clebsch diagonal surface, where all the 27 lines are defined over the field Q, and in particular are all real.
Read more about this topic: Cubic Surface
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