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

2009/10/07 by Nicolas Botbol, Botbol, Nicolas
Mathematics · #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG

paper · pdf · doi:10.48550/arxiv.0910.1340

2 images, 28 pages. To appear in Journal of Pure and Applied Algebra

arxiv created 2010/06/12 · arxiv updated 2010/06/15

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 Pⁿ and (\Bbb P¹)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].

Related

• Implicitization of rational maps
• Images of Rational Maps of Projective Spaces
• The implicitization problem for ϕ: Pⁿ --> (P¹)ⁿ⁺¹
• Hypergraph associahedra and compactifications of moduli spaces of points
• The implicit equation of a multigraded hypersurface