Compactifications of rational maps, and the implicit equations of their images

TL;DR

This paper explores various compactifications of rational maps to describe their images via matrix equations, providing new conditions for such representations without extra hypersurfaces, generalizing previous results.

Contribution

It introduces novel compactifications for rational maps and establishes conditions for matrix representations of their images without additional hypersurfaces.

Findings

Matrix of linear syzygies represents the image closure.

Compactification of the domain as a projective arithmetically Cohen-Macaulay scheme.

Conditions for matrix representations depend on the base locus of the map.

Abstract

In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify $\Bbb {A}^{n-1}$ into an $(n-1)$-dimensional projective arithmetically Cohen-Macaulay subscheme of some $\Bbb {P}^N$. One particular interesting compactification of $\Bbb {A}^{n-1}$ is the toric variety associated to the Newton polytope of the polynomials defining $f$. We consider two different compactifications for the codomain of $f$: $\Bbb {P}^n$ and $(\Bbb {P}^1)^n$. In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established in [BuseJouanolou03], [BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and [Botbol09].