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Tensor product surfaces and quadratic syzygies

Matthew Weaver

TL;DR

This work studies implicitization of tensor product surfaces $X_U$ arising from a basepoint-free 4-dimensional subspace $U$ of $H^0(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(a,b))$ by examining the bigraded ideal $I_U=(p_0,\dots,p_3)$ and its quadratic syzygies. Central to the method is the approximation complex $\mathcal{Z}$, whose bigraded strand $\mathcal{Z}_{2a-1,b-1}$ is controlled by a small, explicitly constructed set of syzygies, once a subspace $V\subseteq U$ is determined from a quadratic syzygy in bidegree $(0,2)$. The paper presents two main scenarios, $\dim V=2$ and $\dim V=3$, and shows that in each case a finite, structured collection of syzygies $Q,S_1,S_2$ (and $S_3$ when $\dim V=3$) suffices to determine the first differential $d_1$ in $\mathcal{Z}_{2a-1,b-1}$; the determinant of this $d_1$-matrix yields a power of the implicit equation $F$ of $X_U$, enabling efficient implicitization beyond full syzygy computations. The results extend prior linear-syzygy phenomena to the quadratic setting and include a running example, discussion of limitations for other bidegrees (notably $(1,1)$), and a conjectural framework for broader $(0,n)$-syzygies. Overall, the approach offers practical, implementable tools for implicitizing tensor product surfaces in CAGD and related applications.

Abstract

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.

Tensor product surfaces and quadratic syzygies

TL;DR

This work studies implicitization of tensor product surfaces arising from a basepoint-free 4-dimensional subspace of by examining the bigraded ideal and its quadratic syzygies. Central to the method is the approximation complex , whose bigraded strand is controlled by a small, explicitly constructed set of syzygies, once a subspace is determined from a quadratic syzygy in bidegree . The paper presents two main scenarios, and , and shows that in each case a finite, structured collection of syzygies (and when ) suffices to determine the first differential in ; the determinant of this -matrix yields a power of the implicit equation of , enabling efficient implicitization beyond full syzygy computations. The results extend prior linear-syzygy phenomena to the quadratic setting and include a running example, discussion of limitations for other bidegrees (notably ), and a conjectural framework for broader -syzygies. Overall, the approach offers practical, implementable tools for implicitizing tensor product surfaces in CAGD and related applications.

Abstract

For a four-dimensional vector space, a basis of defines a rational map . The tensor product surface associated to is the closed image of the map . These surfaces arise within the field of geometric modelling, in which case it is particularly desirable to obtain the implicit equation of . In this paper, we study via the syzygies of the associated bigraded ideal when is free of basepoints, i.e. is regular. Expanding upon work of Duarte and Schenck for such ideals with a linear syzygy, we address the case that has a quadratic syzygy.
Paper Structure (12 sections, 15 theorems, 51 equations)

This paper contains 12 sections, 15 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximation Complex
  4. Multigraded Implicitization
  5. Quadratic Syzygies
  6. Case 1: $\mathop{\rm dim} V=2$
  7. Case 2: $\mathop{\rm dim} V=3$
  8. Syzygies in the case $\mathop{\rm dim} V=2$
  9. Syzygies in the case $\mathop{\rm dim} V=3$
  10. Further observations and questions
  11. Syzygies of bidegree $(1,1)$
  12. Syzygies of bidegree $(0,n)$

Key Result

Theorem 1

Assume that $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ is basepoint free, with $b\geq 3$. Let $I_U$ denote the ideal of $U$, and assume that $I_U$ has a minimal first syzygy $Q$ in bidegree $(0,2)$, and no linear syzygy. Write $V$ to denote the subspace of $U$ associated to Moreover, following Botbol11, the determinant of a $2ab\times 2ab$ matrix representation of $d_1$ i

Theorems & Definitions (41)

  • Theorem
  • Example 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3: Botbol11
  • Lemma 2.4: Botbol11
  • Theorem 2.5: Botbol11
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 31 more