Tensor product surfaces and quadratic syzygies

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-05-06 DOI:10.1016/j.laa.2025.04.020

Matthew Weaver

引用次数: 0

Abstract

For

U \subseteq H^{0} (O_{P^{1} \times 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} : P^{1} \times P^{1} ⇢ 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 [12] for such ideals with a linear syzygy, we address the case that

I_{U}

has a quadratic syzygy.

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张量积曲面与二次合

对于一个四维向量空间U (OP1×P1(a,b))， U的一个基{p0，p1,p2，p3}定义了一个有理映射ϕU:P1×P1与U相关联的张量积曲面是映射 U的封闭图像XU。这些曲面出现在几何建模领域，在这种情况下，特别需要得到XU的隐式方程。在本文中，我们研究了当U没有基点时，即当U是正则的时，通过相关的梯度理想IU=(p0,p1,p2,p3)的合集来研究XU。在Duarte和Schenck[12]对这种理想的线性协同的研究基础上，我们讨论了IU具有二次协同的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.