Matrix Representations of Toric Parametrizations

N. BOTBOL, A. DICKENSTEIN, M. DOHM

COMPUTER AIDED GEOMETRIC DESIGN

ELSEVIER SCIENCE BV

Año: 2009 vol. 26 p. 757 - 771

0167-8396

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