In this article we analyze the implicitization problem of the image of a rational map φ : X P n , with X a toric variety of dimension n − 1 defined by its Cox ring R. Let I := (f 0 , . . . , f n ) be n + 1 homogeneous elements of R. We blow-up the base locus of φ, V (I), and we approximate the Rees algebra Rees R (I) of this blow-up by the symmetric algebra Sym R (I). We provide under suitable assumptions, resolutions Z • for Sym R (I) graded by the divisor group of X , Cl(X ), such that the determinant of a graded strand, det((Z • ) µ ), gives a multiple of the implicit equation, for suitable µ ∈ Cl(X ). Indeed, we compute a region in Cl(X ) which depends on the regularity of Sym R (I) where to choose µ. We also give a geometrical interpretation of the possible other factors appearing in det((Z • ) µ ). A very detailed description is given when X is a multiprojective space.

In this paper we show that a surface in P 3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P 2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P 1 × P 1 in detail and give numerous examples.

In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group.Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal B, obtaining, for each B, a B-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications.We provide tools to transfer knowledge in two directions. First to deduce some information on the graded Betti numbers from the knowledge of regions where the local cohomology with support in a given graded ideal vanishes. This is one of our main results. Conversely, vanishing of local cohomology with support in any graded ideal is deduced from the shifts in a free resolution and the local cohomology of the polynomial ring. Furthermore, the flexibility of treating local cohomology with respect to any B open new possibilities for passing information.We provide new persistence results for the vanishing of local cohomology that extend the fact that weakly regular implies regular in the classical case, and we give sharp estimates for the regularity of a truncation of a module.In the last part, we present a result on Hilbert functions for multigraded polynomial rings, that in particular provides a simple proof of the Grothendieck-Serre formula.

International audienceIn 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

a b s t r a c tIn 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].