On the weak Lefschetz property for ideals generated by powers of general linear forms
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912384560398336 |
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| author | Booth, Matthew D. Singh, Pankaj Vraciu, Adela |
| author_facet | Booth, Matthew D. Singh, Pankaj Vraciu, Adela |
| contents | We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree $d$ holds when the number of variables $n$ is sufficiently large compared to $d$. In particular, we show that if $n\geq 3d-2$ then WLP holds for the ideal generated by squares at the degree $d$ spot and for $n\ge \frac{3d-3}{2}$ WLP holds for ideal generated by cubes at the degree $d$ spot. Finally, we prove that WLP fails for the ideal generated by squares when $n< 3d -2$ at the $d$th spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on $n$ is sharp in the case of the squares. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22542 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the weak Lefschetz property for ideals generated by powers of general linear forms Booth, Matthew D. Singh, Pankaj Vraciu, Adela Commutative Algebra 13D02, 13E10, 13C40 We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree $d$ holds when the number of variables $n$ is sufficiently large compared to $d$. In particular, we show that if $n\geq 3d-2$ then WLP holds for the ideal generated by squares at the degree $d$ spot and for $n\ge \frac{3d-3}{2}$ WLP holds for ideal generated by cubes at the degree $d$ spot. Finally, we prove that WLP fails for the ideal generated by squares when $n< 3d -2$ at the $d$th spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on $n$ is sharp in the case of the squares. |
| title | On the weak Lefschetz property for ideals generated by powers of general linear forms |
| topic | Commutative Algebra 13D02, 13E10, 13C40 |
| url | https://arxiv.org/abs/2410.22542 |