Artículo

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 An-1 into an (n-1)-dimensional projective arithmetically Cohen-Macaulay subscheme of some PN. One particular interesting compactification of An-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: Pn and (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 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]. © 2010 Elsevier B.V.

Registro:

Documento: Artículo
Título:Compactifications of rational maps, and the implicit equations of their images
Autor:Botbol, N.
Filiación:Departamento de Matemática, FCEN, Universidad de Buenos Aires, Argentina
Institut de Mathématiques de Jussieu, Université de P. et M. Curie, Paris VI, France
Año:2011
Volumen:215
Número:5
Página de inicio:1053
Página de fin:1068
DOI: http://dx.doi.org/10.1016/j.jpaa.2010.07.010
Título revista:Journal of Pure and Applied Algebra
Título revista abreviado:J. Pure Appl. Algebra
ISSN:00224049
CODEN:JPAAA
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_00224049_v215_n5_p1053_Botbol.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00224049_v215_n5_p1053_Botbol

Referencias:

  • Adkins, W.A., Hoffman, J.W., Wang, H.H., Equations of parametric surfaces with base points via syzygies (2005) J. Symbolic Comput., 39 (1), pp. 73-101
  • Avramov, L.L., Complete intersections and symmetric algebras (1981) J. Algebra, 73 (1), pp. 248-263
  • (2009), Nicolás Botbol, Code in macaulay2 for computing the toric embedding of (P1)3 in P11; Botbol, N., The implicitization problem for P:Pn→(P1)n+1 (2009) J. Algebra, 322 (11), pp. 3878-3895
  • Botbol, N., Dickenstein, A., Dohm, M., Matrix representations for toric parametrizations (2009) Comput. Aided Geom. Design, 26 (7), pp. 757-771
  • Bruns, W., Gubeladze, J., Viêt Trung Ngô, Normal polytopes, triangulations, and Koszul algebras (1997) J. Reine Angew. Math., 485, pp. 123-160
  • Bruns, W., Herzog, J., Cohen-Macaulay rings (1993) Cambridge Studies in Advanced Mathematics, 39. , Cambridge University Press, Cambridge
  • Busé, L., Chardin, M., Implicitizing rational hypersurfaces using approximation complexes (2005) J. Symbolic Comput., 40 (4-5), pp. 1150-1168
  • Busé, L., Chardin, M., Jouanolou, J.-P., Torsion of the symmetric algebra and implicitization (2009) Proc. Amer. Math. Soc., 137 (6), pp. 1855-1865
  • Busé, L., Cox, D., D'Andrea, C., Implicitization of surfaces in P3 in the presence of base points (2003) J. Algebra Appl., 2 (2), pp. 189-214
  • Busé, L., Dohm, M., Implicitization of bihomogeneous parametrizations of algebraic surfaces via linear syzygies (2007) ISSAC 2007, pp. 69-76. , ACM, New York
  • Busé, L., Jouanolou, J.-P., On the closed image of a rational map and the implicitization problem (2003) J. Algebra, 265 (1), pp. 312-357
  • Marc Chardin, (2004), Regularity of ideals and their powers. Institut de Mathématiques de Jussieu, Prépublication 364; Chardin, M., Implicitization using approximation complexes (2006) Math. Vis., pp. 23-35. , Springer, Berlin, Algebraic Geometry and Geometric Modeling
  • Cox, D., Curves, surfaces, and syzygies (2003) Contemp. Math., 334, pp. 131-150. , Amer. Math. Soc., Providence, RI, Topics in Algebraic Geometry and Geometric Modeling
  • Cox, D., What is a toric variety? (2003) Contemp. Math., 334, pp. 203-223. , Amer. Math. Soc., Providence, RI, Topics in Algebraic Geometry and Geometric Modeling
  • Cox, D.A., Equations of parametric curves and surfaces via syzygies (2001) Contemp. Math., 86, pp. 1-20. , Amer. Math. Soc., Providence, RI, Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000)
  • Dickenstein, A., Feichtner, E.M., Sturmfels, B., Tropical discriminants (2007) J. Amer. Math. Soc., 20 (4), pp. 1111-1133. , (electronic)
  • Gel'fand, I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, resultants, and multidimensional determinants (1994) Mathematics: Theory & Applications, , Birkhäuser Boston Inc., Boston, MA
  • Herzog, J., Simis, A., Vasconcelos, W.V., Koszul homology and blowing-up rings (1983) Lecture Notes in Pure and Appl. Math., 84, pp. 79-169. , Dekker, New York, Commutative Algebra (Trento, 1981)
  • Khetan, A., D'Andrea, C., Implicitization of rational surfaces using toric varieties (2006) J. Algebra, 303 (2), pp. 543-565
  • Knudsen, F.F., Mumford, D., The projectivity of the moduli space of stable curves. I. Preliminaries on "det" and "Div" (1976) Math. Scand., 39 (1), pp. 19-55
  • http://www.math.ucdavis.edu/~latte/, Team Latte, Latte - a software dedicated to the problems of counting and detecting lattice points inside convex polytopes, and the solution of integer programs; Matsumura, H., Commutative ring theory (1989) Cambridge Studies in Advanced Mathematics, 8. , Cambridge University Press, Cambridge, Translated from the Japanese by M. Reid
  • Miller, E., Sturmfels, B., Combinatorial commutative algebra (2005) Graduate Texts in Mathematics, 227. , Springer-Verlag, New York
  • Sederberg, T.W., Chen, F., Implicitization using moving curves and surfaces (1995) SIGGRAPH '95: Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, New York, NY, USA, pp. 301-308. , ACM
  • Vasconcelos, W.V., Arithmetic of blowup algebras (1994) London Mathematical Society Lecture Note Series, 195. , Cambridge University Press, Cambridge

Citas:

---------- APA ----------
(2011) . Compactifications of rational maps, and the implicit equations of their images. Journal of Pure and Applied Algebra, 215(5), 1053-1068.
http://dx.doi.org/10.1016/j.jpaa.2010.07.010
---------- CHICAGO ----------
Botbol, N. "Compactifications of rational maps, and the implicit equations of their images" . Journal of Pure and Applied Algebra 215, no. 5 (2011) : 1053-1068.
http://dx.doi.org/10.1016/j.jpaa.2010.07.010
---------- MLA ----------
Botbol, N. "Compactifications of rational maps, and the implicit equations of their images" . Journal of Pure and Applied Algebra, vol. 215, no. 5, 2011, pp. 1053-1068.
http://dx.doi.org/10.1016/j.jpaa.2010.07.010
---------- VANCOUVER ----------
Botbol, N. Compactifications of rational maps, and the implicit equations of their images. J. Pure Appl. Algebra. 2011;215(5):1053-1068.
http://dx.doi.org/10.1016/j.jpaa.2010.07.010