Tensor product surfaces and quadratic syzygies
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910794779721728 |
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| author | Weaver, Matthew |
| author_facet | Weaver, Matthew |
| contents | For $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ a four-dimensional vector space, a basis $\{p_0,p_1,p_2,p_3\}$ of $U$ defines a rational map $ϕ_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$. The tensor product surface associated to $U$ is the closed image $X_U$ of the map $ϕ_U$. These surfaces arise within the field of geometric modelling, in which case it is particularly desirable to obtain the implicit equation of $X_U$. In this paper, we study $X_U$ via the syzygies of the associated bigraded ideal $I_U=(p_0,p_1,p_2,p_3)$ when $U$ is free of basepoints, i.e. $ϕ_U$ is regular. Expanding upon work of Duarte and Schenck for such ideals with a linear syzygy, we address the case that $I_U$ has a quadratic syzygy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor product surfaces and quadratic syzygies Weaver, Matthew Commutative Algebra 13D02 For $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ a four-dimensional vector space, a basis $\{p_0,p_1,p_2,p_3\}$ of $U$ defines a rational map $ϕ_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$. The tensor product surface associated to $U$ is the closed image $X_U$ of the map $ϕ_U$. These surfaces arise within the field of geometric modelling, in which case it is particularly desirable to obtain the implicit equation of $X_U$. In this paper, we study $X_U$ via the syzygies of the associated bigraded ideal $I_U=(p_0,p_1,p_2,p_3)$ when $U$ is free of basepoints, i.e. $ϕ_U$ is regular. Expanding upon work of Duarte and Schenck for such ideals with a linear syzygy, we address the case that $I_U$ has a quadratic syzygy. |
| title | Tensor product surfaces and quadratic syzygies |
| topic | Commutative Algebra 13D02 |
| url | https://arxiv.org/abs/2501.13032 |