Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T14:51:41.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with μ = 2

Published online by Cambridge University Press:  20 November 2018

Teresa Cortadellas Benítez  and
Carlos D'Andrea
Teresa Cortadellas Benítez
Affiliation:
Universitat de Barcelona, Facultat de Formació del Professorat, 08035 Barcelona, Spain. e-mail: terecortadellas@ub.edu
Carlos D'Andrea
Affiliation:
Universitat de Barcelona, Facultat de Matemàtiques, 08007 Barcelona, Spain. e-mail: cdandrea@ub.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

Keywords

13A3014H50Rees Algebrasrational plane curvesminimal generators
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1225 - 1249
Copyright
Copyright © Canadian Mathematical Society 2014

References

[BH93] Bruns, W. and Herzog, J., Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.Google Scholar
[Bus09] Busé, L., On the equations of the moving curve ideal of a rational algebraic plane curve. J. Algebra. 321(2009), no. 8, 2317–2344.http://dx.doi.org/10.1016/j.jalgebra.2009.01.030 Google Scholar
[BJ03] Busé, L. and Jouanolou, J.-P., On the closed image of a rational map and the implicitization problem. J. Algebra. 265(2003), no. 1, 312–357.http://dx.doi.org/10.1016/S0021-8693(03)00181-9 Google Scholar
[CA00] Casas-Alvero, E., Singularities of plane curves. London Mathematical Society Lecture Note Series, 276, Cambridge University Press, Cambridge, 2000.Google Scholar
[CCL05] Chen, F., Cox, D., and Liu, Y., The μ-basis and implicitization of a rational parametric surface. J. Symbolic Comput. 39(2005), no. 6, 689–706.http://dx.doi.org/10.1016/j.jsc.2005.01.003 Google Scholar
[CWL08] Chen, F., Wang, W. , and Liu, Y., Computing singular points of plane rational curves. J. Symbolic Comput. . 43(2008), no. 2, 92–117.http://dx.doi.org/10.1016/j.jsc.2007.10.003 Google Scholar
[CD10] Cortadellas Benítez, T. and D'Andrea, C., Minimal generators of the defining ideal of the Rees Algebra associated to monoid parametrizations. Comput. Aided Geom. Design. 27(2010), no. 6, 461–473.http://dx.doi.org/10.1016/j.cagd.2010.04.003 Google Scholar
[CD13] Cortadellas Benítez, T. and D'Andrea, C., Rational plane curves parametrizable by conics. J. Algebra. 373(2013) 453–480.http://dx.doi.org/10.1016/j.jalgebra.2012.09.034 Google Scholar
[CD13b] Cortadellas Benítez, T. and D'Andrea, C., Minimal generators of the defining ideal of the Rees Algebra associated to a rational plane parameterization with μ= 2. arxiv:1301.6286 Google Scholar
[Cox08] Cox, D. A., The moving curve ideal and the Rees algebra. Theoret. Comput. Sci. 392(2008), no. 1–3, 23–36.http://dx.doi.org/10.1016/j.tcs.2007.10.012 Google Scholar
[CGZ00] Cox, D., Goldman, R., and Zhang, M. , On the validity of implicitization by moving quadrics of rational surfaces with no base points. J. Symbolic Comput. 29(2000), no. 3, 419–440.http://dx.doi.org/10.1006/jsco.1999.0325 Google Scholar
[CHW08] Cox, D., Hoffman, J. W., and Wang, H., Syzygies and the Rees algebra. J. Pure Appl. Algebra. 212(2008), no. 7, 1787–1796.http://dx.doi.org/10.1016/j.jpaa.2007.11.006 Google Scholar
[CKPU11] Cox, D., Kustin, A., Polini, C., and Ulrich, B., A study of singularities on rational curves via syzygies. Mem. Amer. Math. Soc. 222(2013), no. 1045.Google Scholar
[CLO07] Cox, D., Little, J., and O'Shea, D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Third ed., Undergraduate Texts in Mathematics, Springer, New York, 2007.Google Scholar
[CSC98] Cox, D. A., Sederberg, T. W., and Chen, F., The moving line ideal basis of planar rational curves. Comput. Aided Geom. Design. 15(1998), no. 8, 803–827.http://dx.doi.org/10.1016/S0167-8396(98)00014-4 Google Scholar
[Mac] Grayson, D. R. and Stillman, M. E., Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/.Google Scholar
[HS12] Hassanzadeh, S. H. and Simis, A. , Implicitization of the Jonquières parametrizations. arxiv:1205.1083.Google Scholar
[HSV08] Hong, J., Simis, A., and Vasconcelos, W. V., On the homology of two-dimensional elimination. J. Symbolic Comput. 43(2008), no. 4, 275–292.http://dx.doi.org/10.1016/j.jsc.2007.10.010 Google Scholar
[HSV09] Hong, J., Simis, A., and Vasconcelos, W. V., The equations of almost complete intersections. Bull. Braz. Math. Soc. 43(2012), no. 2, 171–199.Google Scholar
[HW10] Hoffman, J. W. and Wang, H., Defining equations of the Rees algebra of certain parametric surfaces. J. Algebra Appl. 9(2010), no. 6, 1033–1049.http://dx.doi.org/10.1142/S0219498810004385 Google Scholar
[Jou97] Jouanolou, J. P., Formes d'inertie et résultant: un formulaire. Adv. Math. 126(1997), no. 2, 119–250.http://dx.doi.org/10.1006/aima.1996.1609 Google Scholar
[KPU09] Kustin, A. R., Polini, C., and Ulrich, B., Rational normal scrolls and the defining equations of Rees algebras. J. Reine Angew. Math. 650(2011), 23–65.Google Scholar
[KPU13] Kustin, A. R., Polini, C., and Ulrich, B., The bi-graded structure of symmetric algebras with applications to Rees rings. arxiv:1301.7106 Google Scholar
[SC95] Sederberg, T. and Chen, F., Implicitization using moving curves and surfaces. Proceedings of SIGGRAPH. 1995, 301–308.Google Scholar
[SGD97] Sederberg, T., Goldman, R., and Du, H., Implicitizing rational curves by the method of moving algebraic curves. In: Parametric algebraic curves and applications (Albuquerque, NM, 1995) J. Symbolic Comput. 23(1997), no. 2–3, 153–175.http://dx.doi.org/10.1006/jsco.1996.0081 Google Scholar
[SWP08] Sendra, J. R., Winkler, F., and Pérez-Díaz, S., Rational algebraic curves. A computer algebra approach. Algorithms and Computation in Mathematics, 22, Springer, Berlin, 2008.Google Scholar
[Wol10] Wolfram Research, Inc. Mathematica, Version 8.0, Champaign, IL, 2010.Google Scholar
[ZCG99] Zhang, M., Chionh, E.-W., and Goldman, R. N., On a relationship between the moving line and moving conic coefficient matrices. Comput. Aided Geom. Design 16(1999), no. 6, 517–527.http://dx.doi.org/10.1016/S0167-8396(99)00014-X Google Scholar