Rational Normal Curves, Chip Firing and Free Resolutions

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.