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

TL;DR

This paper extends known results about the generators and relations of the ring of invariants for n points on a line, showing they hold integrally over Z[1/12!] and possibly over Z[1/6], with implications for algebraic geometry.

Contribution

It proves that the generators and relations of the invariant ring are integral over Z[1/12!], strengthening previous rational results and suggesting broader integrality properties.

Findings

Generators are in degree one, as shown by Kempe.

Relations are generated by quadratic binomials, except for n=6 where a cubic appears.

Results hold over Z[1/12!] and potentially over Z[1/6].

Abstract

Consider the projective coordinate ring of the GIT quotient (P^1)^n//SL(2), with the usual linearization, where n is even. In 1894, Kempe proved that this ring is generated in degree one. In [HMSV2] we showed that, over the rationals, the relations between degree one invariants are generated by a class of quadratic relations -- the simplest binomial relations -- with the exception of n=6, where there is a single cubic relation. The purpose of this paper is to show that these results hold over Z[1/12!], and to suggest why they may be true over Z[1/6].