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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.02135 |
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| _version_ | 1866913585447305216 |
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| author | Huang, Daoji Larson, Matt |
| author_facet | Huang, Daoji Larson, Matt |
| contents | We give a criterion for a collection of polynomials to be a universal Gröbner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give simple proofs of several existing results on universal Gröbner bases. We introduce fine Schubert polynomials, which record the multidegrees of the closures of matrix Schubert varieties in $(\mathbb{P}^1)^{n^2}$. We compute the fine Schubert polynomials of permutations $w$ where the coefficients of the Schubert polynomials of $w$ and $w^{-1}$ are all either 0 or 1, and we use this to give a universal Gröbner basis for the ideal of the matrix Schubert variety of such a permutation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fine multidegrees, universal Grobner bases, and matrix Schubert varieties Huang, Daoji Larson, Matt Algebraic Geometry Commutative Algebra Combinatorics 14M15, 13P10, 13C40, 14M12, 05E14 We give a criterion for a collection of polynomials to be a universal Gröbner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give simple proofs of several existing results on universal Gröbner bases. We introduce fine Schubert polynomials, which record the multidegrees of the closures of matrix Schubert varieties in $(\mathbb{P}^1)^{n^2}$. We compute the fine Schubert polynomials of permutations $w$ where the coefficients of the Schubert polynomials of $w$ and $w^{-1}$ are all either 0 or 1, and we use this to give a universal Gröbner basis for the ideal of the matrix Schubert variety of such a permutation. |
| title | Fine multidegrees, universal Grobner bases, and matrix Schubert varieties |
| topic | Algebraic Geometry Commutative Algebra Combinatorics 14M15, 13P10, 13C40, 14M12, 05E14 |
| url | https://arxiv.org/abs/2410.02135 |