Such forms F, and the
hypersurfacesF = 0 they define in
projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the
Fermat curves, and other examples well known in the theory of
Diophantine equations.
gives the Fermat
cubic surface in P3 with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (x : ax : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.