Rational Normal Curves, Chip Firing and Free Resolutions

Rahul Karki; Madhusudan Manjunath

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arXiv · 2301.09104

Rational Normal Curves, Chip Firing and Free Resolutions

Rahul Karki·Madhusudan Manjunath

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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.

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BibTeX RIS

Rahul Karki, Madhusudan Manjunath. 2023-01-22. Rational Normal Curves, Chip Firing and Free Resolutions. https://arxiv.org/abs/2301.09104

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