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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.14756 |
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| _version_ | 1866913359891267584 |
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| author | Mezzetti, Emilia Miró-Roig, Rosa M. |
| author_facet | Mezzetti, Emilia Miró-Roig, Rosa M. |
| contents | In this paper, we determine the maximum $h_{max}$ and the minimum $h_{min}$ of the Hilbert vectors of Perazzo algebras $A_F$, where $F$ is a Perazzo polynomial of degree $d$ in $n+m+1$ variables. These algebras always fail the Strong Lefschetz Property. We determine the integers $n,m,d$ such that $h_{max}$ (resp. $h_{min}$) is unimodal, and we prove that $A_F$ always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in $\mathbb P^4$ with minimum Hilbert vectors. Finally we pose some open problems in this context.
Dedicated to Enrique Arrondo on the occasion of his $60^{th}$ birthday. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perazzo $n$-folds and the weak Lefschetz property Mezzetti, Emilia Miró-Roig, Rosa M. Commutative Algebra Algebraic Geometry 14J70, 14M05, 13E10 In this paper, we determine the maximum $h_{max}$ and the minimum $h_{min}$ of the Hilbert vectors of Perazzo algebras $A_F$, where $F$ is a Perazzo polynomial of degree $d$ in $n+m+1$ variables. These algebras always fail the Strong Lefschetz Property. We determine the integers $n,m,d$ such that $h_{max}$ (resp. $h_{min}$) is unimodal, and we prove that $A_F$ always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in $\mathbb P^4$ with minimum Hilbert vectors. Finally we pose some open problems in this context. Dedicated to Enrique Arrondo on the occasion of his $60^{th}$ birthday. |
| title | Perazzo $n$-folds and the weak Lefschetz property |
| topic | Commutative Algebra Algebraic Geometry 14J70, 14M05, 13E10 |
| url | https://arxiv.org/abs/2405.14756 |