On virtual resolutions of points in a product of projective spaces
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911780065771520 |
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| author | Bailly-Hall, Isidora Berkesch, Christine Dovgodko, Karina Guan, Sean Sivakumar, Saisudharshan Sun, Jishi |
| author_facet | Bailly-Hall, Isidora Berkesch, Christine Dovgodko, Karina Guan, Sean Sivakumar, Saisudharshan Sun, Jishi |
| contents | For finite sets of points in $\mathbb{P}^n \times \mathbb{P}^m$, we produce short virtual resolutions, as introduced by Berkesch--Erman--Smith. We first intersect with a sufficiently high power of one set of variables for points in $\mathbb{P}^n \times \mathbb{P}^m$ to produce a virtual resolution of length $n+m$. Then, we describe an explicit virtual resolution of length 3 for a set of points in sufficiently general position in $\mathbb{P}^1 \times \mathbb{P}^2$, via a subcomplex of a free resolution. This first result generalizes to $\mathbb{P}^n \times \mathbb{P}^m$ work of Harada--Nowroozi--Van Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl and Booms-Peot, which were both for $\mathbb{P}^1 \times \mathbb{P}^1$. Along the way, we also note an explicit relationship between Betti numbers and higher difference matrices of bigraded Hilbert functions for $\mathbb{P}^n \times \mathbb{P}^m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On virtual resolutions of points in a product of projective spaces Bailly-Hall, Isidora Berkesch, Christine Dovgodko, Karina Guan, Sean Sivakumar, Saisudharshan Sun, Jishi Commutative Algebra Algebraic Geometry Primary: 13D02. Secondary: 14M25, 14F06 For finite sets of points in $\mathbb{P}^n \times \mathbb{P}^m$, we produce short virtual resolutions, as introduced by Berkesch--Erman--Smith. We first intersect with a sufficiently high power of one set of variables for points in $\mathbb{P}^n \times \mathbb{P}^m$ to produce a virtual resolution of length $n+m$. Then, we describe an explicit virtual resolution of length 3 for a set of points in sufficiently general position in $\mathbb{P}^1 \times \mathbb{P}^2$, via a subcomplex of a free resolution. This first result generalizes to $\mathbb{P}^n \times \mathbb{P}^m$ work of Harada--Nowroozi--Van Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl and Booms-Peot, which were both for $\mathbb{P}^1 \times \mathbb{P}^1$. Along the way, we also note an explicit relationship between Betti numbers and higher difference matrices of bigraded Hilbert functions for $\mathbb{P}^n \times \mathbb{P}^m$. |
| title | On virtual resolutions of points in a product of projective spaces |
| topic | Commutative Algebra Algebraic Geometry Primary: 13D02. Secondary: 14M25, 14F06 |
| url | https://arxiv.org/abs/2402.12495 |