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In the early 1870s, one of the leading Mathematicians of the time, Alfred Clebsch, constructed his famous Diagonal Cubic Surface. It is well-known because all the 27 straight lines which are contained in any cubic surface, are very symmetric and visible in the real part of the surface.
But he did not only construct this very special example of a cubic surface, but he also proved many interesting results on cubic surfaces. A particularily interesting one is the fact that for each such surface it is possible to write down - in terms of certain determinants - the equation of some other surface of degree 9 which cuts the original one exactly in all its straight lines (notice that 3*9=27). On this model we show the cubic surface together with its 27 straight lines and also a small part of the so-called covariant of degree 9 (in blue).