Tensor product surfaces and graded syzygies
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913103685353472 |
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| author | Weaver, Matthew |
| author_facet | Weaver, Matthew |
| contents | Let $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ be a four-dimensional vector space and consider the rational map $ϕ_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$ defined by its basis of bihomogeneous polynomials. The tensor product surface $X_U\subseteq \mathbb{P}^3$ is the closed image of $ϕ_U$, and a fundamental problem in this setting is to determine its implicit equation. As these surfaces are ubiquitous within the field of geometric modeling and design, knowledge of their implicit equations is particularly advantageous, allowing for more effective and efficient computations. In this article, we expand upon work of Duarte-Schenck and work of the present author to solve this implicitization problem when the bigraded ideal $I_U$ admits a singly graded syzygy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07974 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tensor product surfaces and graded syzygies Weaver, Matthew Algebraic Geometry Commutative Algebra 14Q10, 13D02 Let $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ be a four-dimensional vector space and consider the rational map $ϕ_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$ defined by its basis of bihomogeneous polynomials. The tensor product surface $X_U\subseteq \mathbb{P}^3$ is the closed image of $ϕ_U$, and a fundamental problem in this setting is to determine its implicit equation. As these surfaces are ubiquitous within the field of geometric modeling and design, knowledge of their implicit equations is particularly advantageous, allowing for more effective and efficient computations. In this article, we expand upon work of Duarte-Schenck and work of the present author to solve this implicitization problem when the bigraded ideal $I_U$ admits a singly graded syzygy. |
| title | Tensor product surfaces and graded syzygies |
| topic | Algebraic Geometry Commutative Algebra 14Q10, 13D02 |
| url | https://arxiv.org/abs/2605.07974 |