On the X-rank with respect to linearly normal curves

• Autores: Edoardo Ballico, Alessandra Bernardi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 2, 2013, págs. 141-154
• Idioma: inglés
• DOI: 10.1007/s13348-011-0058-4
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper we study the X-rank of points with respect to smooth linearly normal curves {X \subset \mathbb {P}^{n}} of genus g and degree n+g. We prove that, for such a curve X, under certain circumstances, the X-rank of a general point of X-border rank equal to s is less or equal than n + 1 − s. In the particular case of g = 2 we give a complete description of the X-rank if n = 3, 4; while if n ≥ 5 we study the X-rank of points belonging to the tangential variety of X.

• Referencias bibliográficas
  • Ådlansdvik B.: Joins and higher secant varieties. Math. Scand. 61, 213–222 (1987)
  • Allman S., Rhodes J.: Phylogenetic ideals and varieties for the general Markov model. Adv. Appl. Math. 40(2), 127–148 (2008)
  • Arbarello, E., Cornalba, M., Griffiths, P.A., Harris J.: Geometry of algebraic curves, vol. I. Grundlehren der Mathematischen Wissenschaften,...
  • Ballico, E., Bernardi, A.: On the X-rank with respect to linear projections of projective varieties. (full text) (arxiv version), Math. Nachr....
  • ten Berge J.M.F., Castaing J., Comon P., Lathauwer L.: Generic and typical ranks of multi-way arrays. Linear Algebra Appl. 430(11–12), 2997–3007...
  • Bernardi A.: Ideals of varieties parameterizing certain symmetric tensors. J. P. A. Algebra 212(6), 1542–1559 (2008)
  • Bernardi, A., Gimigliano, A., Idà, M.: Computing symmetric rank for symmetric tensors. (full text) (arxiv version), J. Symbolic Comput. 46,...
  • Buczynski, J., Landsberg J., M.: Ranks of tensors and a generalization of secant varieties. http://arxiv.org/abs/0909.4262, (2009, in press)
  • Catalisano, M.V., Geramita, A.V., Gimigliano, A.: On the rank of tensors, via secant varieties and fat points, zero-dimensional schemes and...
  • Catalisano M.V., Geramita A.V., Gimigliano A.: Ranks of tensors, secant varieties of Segre varieties and fat points. Linear Algebra Appl....
  • Catalisano M.V., Geramita A.V., Gimigliano A.: On the ideals of secant varieties to certain rational varieties. J. Algebra 319, 1913–1931...
  • Comas, G., Seiguer, M.: On the rank of a binary form. 11(1), 65–78. doi:10.1007/s10208-010-9077-x.
  • Comon P., Golub G., Lim L.H., Mourrain B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30, 1254–1279 (2008)
  • Comon, P., Ottaviani, G.: On the typical rank of real binary forms. http://arxiv.org/abs/math/0909.4865 (2009, in press)
  • Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985),...
  • Friedland, S.: On the generic rank of 3-tensors. http://arxiv.org/abs/math/0805.3777 (2008, in press)
  • Landsberg J.M., Manivel L.: On the ideals of secant varieties of Segre varieties. Found. Comput. Math. 4(4), 397–422 (2004)
  • Landsberg J.M., Manivel L.: Generalizations of Strassen’s equations for Secant varieties of Segre varieties. Commun. Algebra 36, 1–18 (2008)
  • Landsberg, J.M., Teiler, Z.: On the ranks and border ranks of symmetric tensors. http://arxiv.org/abs/0901.0487v3 (2009, in press)
  • Landsberg J.M., Weyman J.: On the ideals and singularities of secant varieties of Segre varieties. Bull. Lond. Math. Soc. 39(4), 685–697 (2007)
  • Lim L.H., de Silva V.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127...
  • Manivel L.: On spinor varieties and their Secants. SIGMA, Contribution to the Special Issue lie Cartan and Differential Geometry 5, 78–100...
  • Piene R.: Cuspidal projections of space curves. Math. Ann. 256(1), 95–119 (1981)
  • Strassen V.: Rank and optimal computation of generic tensors. Linear Algebra Appl. 52/53, 645–685 (1983)
  • Sylvester J.: Sur une extension d’un théorème de Clebsh relatif aux courbes du quatrième degré. Comptes Rendus, Math. Acad. Sci. Paris 102,...