Matrix representations for toric parametrizations

Authors:
Nicolás Botbol;Alicia Dickenstein;Marc Dohm
Affiliations:
Departamento de Matemática, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pab.I, 1428 Buenos Aires, Argentina and Institut de Mathématiques de Jussieu, Université de P. e ...;Departamento de Matemática, FCEN, Universidad de Buenos Aires, Ciudad Universitaria, Pab.I, 1428 Buenos Aires, Argentina;Laboratoire J. A. Dieudonné, Université de Nice -- Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 2, France
Venue:
Computer Aided Geometric Design
Year:
2009

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Abstract

In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [Buse, L., Jouanolou, J.-P., 2003. J. Algebra 265 (1), 312-357] and [Buse, L., Chardin, M.J., 2005. Symbolic Comput. 40 (4-5), 1150-1168]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T=P^1xP^1 in detail and give numerous examples.