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 A(n-1) into an (n - 1)-dimensional projective arithmetically Cohen-Macaulay subscheme of some P-N. One particular interesting compactification of 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: P-n and (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 by Laurent Buse and Jean-Pierre jouanolou (2003) [12], Laurent Buse et al. (2009)[9], Laurent Buse and Marc Dohm (2007) [11], Nicolas Botbol et al. (2009) [5] and Nicolas Botbol (2009) [4]. (C) 2010 Elsevier B.V. All rights reserved.