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

Abstract

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 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].