Symbolic powers of the generic linkage of maximal minors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908971623776256 |
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| author | Pandey, Vaibhav Varbaro, Matteo |
| author_facet | Pandey, Vaibhav Varbaro, Matteo |
| contents | Let $I$ be the ideal generated by the maximal minors of a matrix of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit description of the lead terms of the generators of $J$ using Gröbner degeneration. Indeed, we construct a degeneration which preserves the entire graded Betti table of $J$ on passing to the initial ideal. We leverage this construction to establish the equality of the symbolic and ordinary powers of $J$. Our analysis of the initial ideal readily yields the Gorenstein property of the associated graded ring of $J$, and, in positive characteristic, the $F$-rationality of the Rees algebra of $J$. Using the technique of $F$-split filtrations, we further obtain the $F$-regularity of the blowup algebras of $J$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11235 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symbolic powers of the generic linkage of maximal minors Pandey, Vaibhav Varbaro, Matteo Commutative Algebra Algebraic Geometry 13C40, 13A35, 13A30, 14M06 (Primary) 14M10 ( Secondary) Let $I$ be the ideal generated by the maximal minors of a matrix of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit description of the lead terms of the generators of $J$ using Gröbner degeneration. Indeed, we construct a degeneration which preserves the entire graded Betti table of $J$ on passing to the initial ideal. We leverage this construction to establish the equality of the symbolic and ordinary powers of $J$. Our analysis of the initial ideal readily yields the Gorenstein property of the associated graded ring of $J$, and, in positive characteristic, the $F$-rationality of the Rees algebra of $J$. Using the technique of $F$-split filtrations, we further obtain the $F$-regularity of the blowup algebras of $J$. |
| title | Symbolic powers of the generic linkage of maximal minors |
| topic | Commutative Algebra Algebraic Geometry 13C40, 13A35, 13A30, 14M06 (Primary) 14M10 ( Secondary) |
| url | https://arxiv.org/abs/2412.11235 |