Rational Normal Curves, Chip Firing and Free Resolutions
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913581552893952 |
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| author | Karki, Rahul Manjunath, Madhusudan |
| author_facet | Karki, Rahul Manjunath, Madhusudan |
| contents | 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 Gröbner degeneration and an explicit combinatorial minimal free resolution of this Gröbner 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09104 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rational Normal Curves, Chip Firing and Free Resolutions Karki, Rahul Manjunath, Madhusudan Commutative Algebra Algebraic Geometry Combinatorics Rings and Algebras 14M12, 13D02, 05E40, 05E14, 05C57, 52C35 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 Gröbner degeneration and an explicit combinatorial minimal free resolution of this Gröbner 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. |
| title | Rational Normal Curves, Chip Firing and Free Resolutions |
| topic | Commutative Algebra Algebraic Geometry Combinatorics Rings and Algebras 14M12, 13D02, 05E40, 05E14, 05C57, 52C35 |
| url | https://arxiv.org/abs/2301.09104 |