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We consider the ring of invariants of n points on the projective line. Thespace (P^1)^n // PGL_2 is perhaps the first nontrivial example of a GeometryInvariant Theory quotient. The construction depends on the weighting of the npoints. Kempe discovered a beautiful set of generators (at least in the case ofunit weights) in 1894. We describe the full ideal of relations for all possibleweightings. In some sense, there is only one equation, which is quadric exceptfor the classical case of the Segre cubic primal, for n=6 and weight 1^6. Thecases of up to 6 points are long known to relate to beautiful familiargeometry. The case of 8 points turns out to be richer still.

The relations among invariants of points on the projective line