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

In this paper, we give different compactifications for the domain and the codomain of an affine rational map f which parameterizes 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 into an (n−1)-dimensional projective arithmetically Cohen–Macaulay subscheme of some . One particular interesting compactification of is the toric variety associated to the Newton polytope of the polynomials defining f. We consider two different compactifications for the codomain of f: and . 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 by Laurent Busé and Jean-Pierre Jouanolou (2003) [12], , Laurent Busé et al. (2009) [9], , Laurent Busé and Marc Dohm (2007) [11], , Nicolás Botbol et al. (2009) [5], and Nicolás Botbol (2009) [4].