Matrix representations for toric parametrizations

TL;DR

This paper develops a matrix-based method to represent surfaces parametrized over toric varieties, generalizing previous results and leveraging sparsity for smaller matrices, applicable to complex parametrizations.

Contribution

It extends matrix representation techniques for surface parametrizations from P^2 to general toric varieties, including P^1 x P^1, improving efficiency and applicability.

Findings

Smaller matrices due to exploiting sparsity

Generalization from P^2 to toric varieties

Effective for parametrizations with finite base points

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 [BJ03] and [BC05]. 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^1 x P^1 in detail and give numerous examples.