A package for computing implicit equations of parametrizations from toric surfaces

TL;DR

This paper introduces an algorithm implemented in Macaulay2 for computing matrix representations of surfaces in P^3 parametrized over toric varieties, leveraging sparsity for efficiency.

Contribution

It extends previous methods to handle more general base loci and exploits sparsity to produce smaller matrices for implicitization.

Findings

Successfully implemented in Macaulay2.

Handles non-complete intersection base loci.

Produces smaller matrices by exploiting sparsity.

Abstract

In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in Macaulay2. We showed in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Botbol-09] we generalized this to the case where the base locus is not necessarily a local complete intersection. The key point consists in exploiting the sparse structure of the parametrization, which allows us to obtain significantly smaller matrices than in the homogeneous case.