The Betti Numbers of Kunz-Waldi Semigroups
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912521813753856 |
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| author | González-Sánchez, Mario Singh, Srishti Srinivasan, Hema |
| author_facet | González-Sánchez, Mario Singh, Srishti Srinivasan, Hema |
| contents | Given two coprime numbers $p<q$, KW semigroups contain $p,q$ and are contained in $\langle p,q,r \rangle$ where $2r= p,q, p+q$ whichever is even. These semigroups were first introduced by Kunz and Waldi. Kunz and Waldi proved that all $KW$ semigroups of embedding dimension $n\geq 4$ have Cohen-Macaulay type $n-1$ and first Betti number ${n \choose 2}$. In this paper, we characterize KW semigroups whose defining ideal is generated by the $2\times 2$ minors of a $2\times n$ matrix. In addition, we identify all KW semigroups that lie on the interior of the same face of the Kunz cone $\mathcal C_p$ as a KW semigroup with determinantal defining ideal. Thus, we provide an explicit formula for the Betti numbers of all those KW semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Betti Numbers of Kunz-Waldi Semigroups González-Sánchez, Mario Singh, Srishti Srinivasan, Hema Commutative Algebra 13D02, 13D05 (Primary), 20M14 (Secondary) Given two coprime numbers $p<q$, KW semigroups contain $p,q$ and are contained in $\langle p,q,r \rangle$ where $2r= p,q, p+q$ whichever is even. These semigroups were first introduced by Kunz and Waldi. Kunz and Waldi proved that all $KW$ semigroups of embedding dimension $n\geq 4$ have Cohen-Macaulay type $n-1$ and first Betti number ${n \choose 2}$. In this paper, we characterize KW semigroups whose defining ideal is generated by the $2\times 2$ minors of a $2\times n$ matrix. In addition, we identify all KW semigroups that lie on the interior of the same face of the Kunz cone $\mathcal C_p$ as a KW semigroup with determinantal defining ideal. Thus, we provide an explicit formula for the Betti numbers of all those KW semigroups. |
| title | The Betti Numbers of Kunz-Waldi Semigroups |
| topic | Commutative Algebra 13D02, 13D05 (Primary), 20M14 (Secondary) |
| url | https://arxiv.org/abs/2503.16736 |