The ring of evenly weighted points on the line

TL;DR

This paper provides a formula for the Hilbert function of a GIT quotient related to evenly weighted points on a line, revealing algebraic properties like Koszulness and connecting to Kostka numbers.

Contribution

It introduces a closed formula for the Hilbert function of the coordinate ring of the quotient, and shows that for even weights, the ring has a quadratic Gr"obner basis and is Koszul.

Findings

Computed the degree of the GIT quotient.

Established the Koszul property for even weights.

Connected the Hilbert function to stretched Kostka numbers.

Abstract

Let $M_w = (\Pj^1)^n \q \mathrm{SL}_2$ denote the geometric invariant theory quotient of $(\Pj^1)^n$ by the diagonal action of $\mathrm{SL}_2$ using the line bundle $\mathcal{O}(w_1,w_2,...,w_n)$ on $(\Pj^1)^n$. Let $R_w$ be the coordinate ring of $M_w$. We give a closed formula for the Hilbert function of $R_w$, which allows us to compute the degree of $M_w$. The graded parts of $R_w$ are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights $w_i$ are even, we find a presentation of $R_w$ so that the ideal $I$ of this presentation has a quadratic Gr\"obner basis. In particular, $R_w$ is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of $M_w$.