Effective criteria for bigraded birational maps

TL;DR

This paper develops new matrix-based criteria for determining when rational maps from products of subvarieties, especially low bidegree maps from two lines to a plane, are birational, with applications to geometric modeling.

Contribution

It introduces novel birationality criteria using Jacobian dual matrices and syzygy analysis for maps from products of varieties, especially in low bidegrees.

Findings

New criteria for birationality based on matrix ranks.

Characterization of birational maps from two lines to the plane.

Applications to geometric modeling contexts.

Abstract

In this paper, we consider rational maps whose source is a product of two subvarieties, each one being embedded in a projective space. Our main objective is to investigate birationality criteria for such maps. First, a general criterion is given in terms of the rank of a couple of matrices that became to be known as Jacobian dual matrices. Then, we focus on rational maps from the product of two projectine lines to the projective plane in very low bidegrees and provide new matrix-based birationality criteria by analyzing the syzygies of the defining equations of the map, in particular by looking at the dimension of certain bigraded parts of the syzygy module. Finally, applications of our results to the context of geometric modeling are discussed at the end of the paper.