Rahul Karki | Searcharxiv

arXiv2026

Restricted Chip-Firing: Toric Toppling Ideals, Picard Groups, and Cellular Resolutions

We study certain groups and ideals arising from the chip-firing game on a generalisation of graphs called pargraphs. Several well-known families of toric ideals arise as toppling ideals of pargraphs. These include ideals defining rational normal curves, binomial edge ideals of complete graphs, and toric ideals of certain Fano polytopes. As our central result, we provide several equivalent conditions for the Picard group of a pargraph to be free, which equivalently characterise when its toppling ideal is toric. Furthermore, we construct a Gr\"obner basis for the toppling ideal, a minimal cellular free resolution for a distinguished initial ideal known as the $G$-parking function ideal, and establish Cohen-Macaulay property for these ideals.

math.AC↗

arXiv2023

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.

math.AC↗