An algorithm for computing implicit equations of bigraded rational surfaces

TL;DR

This paper presents a practical algorithm implemented in Macaulay2 for computing implicit equations of bigraded rational surfaces, extending theoretical methods to concrete examples and aiding understanding of the underlying algebraic geometry.

Contribution

It adapts and implements an existing theoretical method for implicitization of bigraded surfaces in a computer algebra system, facilitating practical computations.

Findings

Successfully computes implicit equations for bigraded surfaces

Provides small examples illustrating the theory

Complements existing algorithms with practical implementation

Abstract

In this article we show how to compute a matrix representation and the implicit equation by means of the method developed in [Botbol: arXiv:1007.3437], using the computer algebra system Macaulay2 \cite{M2}. As it is probably the most interesting case from a practical point of view, we restrict our computations to parametrizations of bigraded surfaces. This implementation allows to compute small examples for the better understanding of the theory developed in [Botbol: arXiv:1007.3437], and is a complement to the algorithm [Botbol, Dohm: arXiv:1001.1126].