The relations among invariants of points on the projective line

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