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On the Clebsch Diagonal Cubic, all 27 of the complex lines present on a general smooth Cubic Surface are real. In addition, there are 10 points on the surface where three of the 27 lines meet. These points are called Eckardt points (Fischer 1986).
See also Clebsch Diagonal Cubic, Cubic Surface
References
Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums.
Braunschweig, Germany: Vieweg, p. 11, 1986.