Tensor product surfaces and quadratic syzygies

2025/01/22 by Matthew D. Weaver, Weaver, Matthew
Computer Science · Mathematics · #13D02 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Mathematics and Applications #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2501.13032

openalex publication_date 2025/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For U⊆ H⁰(O_{ℙ¹× ℙ¹}(a,b)) a four-dimensional vector space, a basis \p₀,p₁,p₂,p₃\ of U defines a rational map ϕ_U: ℙ¹× ℙ¹ \dashrightarrow ℙ³. 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=(p₀,p₁,p₂,p₃) 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.

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