Matrix representations for toric parametrizations

In this paper we show that a surface in P3 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 P2 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 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=P1×P1 in detail and give numerous examples.