The ideal of relations for the ring of invariants of n points on the line

TL;DR

This paper characterizes the relations among invariants of n points on a line, revealing a simple quadratic relation for most cases and a classical relation for six points, completing a longstanding research program.

Contribution

It provides a comprehensive description of the relations among invariants for all weights, including a single quadratic relation in the main case and relations inherited from 8 points.

Findings

For n ≠ 6, the relation is a quadratic binomial.

For n=6, the relation is the classical Segre cubic.

The ideal of relations is generated by quadratics from the 8-point case.

Abstract

The study of the projective coordinate ring of the (geometric invariant theory) moduli space of n ordered points on P^1 up to automorphisms began with Kempe in 1894, who proved that the ring is generated in degree one in the main (n even, unit weight) case. We describe the relations among the invariants for all possible weights. In the main case, we show that up to the symmetric group symmetry, there is a single equation. For n not 6, it is a simple quadratic binomial relation. (For n=6, it is the classical Segre cubic relation.) For general weights, the ideal of relations is generated by quadratics inherited from the case of 8 points. This paper completes the program set out in [HMSV1].