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for some given degree m, summed for 1 ≤ i ≤ n.
Such forms F, and the hypersurfaces F = 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.
- is the unit circle in P2
- is the unit hyperbola in P2.
- 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.
- gives a K3 surface in P3.