Rational Normal Curves, Chip Firing and Free Resolutions

TL;DR

This paper links rational normal curves to chip firing games, using combinatorial methods to study their ideals, resolutions, and Hilbert series, providing explicit constructions and new insights into their algebraic structure.

Contribution

It introduces a novel combinatorial approach connecting rational normal curves with generalized cycle graphs, enabling explicit resolutions and formulas.

Findings

Constructed explicit Gr"obner degenerations for Cohen-Macaulay initial ideals.

Provided combinatorial formulas for Hilbert series of lex-segment ideals.

Developed a new perspective on the Eagon-Northcott complex for rational normal curves.

Abstract

We study rational normal curves via a connection to the chip firing game. A key technique, introduced in this article, is to interpret the defining ideal of the rational normal curve as an ideal associated to a generalisation of a cycle graph called a parcycle. This association allows us to study rational normal curves by combinatorial methods. Given any Cohen-Macaulay initial monomial ideal of the rational normal curve, we explicitly construct (via this association) a corresponding Gr\"obner degeneration and an explicit combinatorial minimal free resolution of this Gr\"obner degeneration. Applications include minimal cellular resolutions for each Cohen-Macaulay initial monomial ideal of the rational normal curve, explicit combinatorial formulas for Hilbert series of certain lex-segment ideals and a combinatorial perspective on the Eagon-Northcott complex associated to the rational normal curve.