Restricted secant varieties of Grassmannians

• Bidlema, Dalton ^[1] ; Oeding, Luke ^[1]
  1. [1] Auburn University

     Auburn University

     Estados Unidos

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 2, 2024, págs. 545-565
• Idioma: inglés
• DOI: 10.1007/s13348-023-00399-4
• Texto completo no disponible (Saber más ...)
• Resumen

  • Restricted secant varieties of Grassmannians are constructed from sums of points corresponding to k-planes with the restriction that their intersection has a prescribed dimension. We study dimensions of restricted secant of Grassmannians and relate them to the analogous question for secants of Grassmannians via an incidence variety construction. We define a notion of expected dimension and give a formula for the dimension of all restricted secant varieties of Grassmannians that holds if the BDdG conjecture [Baur et al. in Exp Math 16(2):239–250, 2007, Conjecture 4.1] on non-defectivity of Grassmannians is true. We also demonstrate example calculations in Macaulay 2, and point out ways to make these calculations more efficient. We also show a potential application to coding theory.

• Referencias bibliográficas
  • Abo, H., Ottaviani, G., Peterson, C.: Non-defectivity of Grassmannians of planes. J. Algebraic Geom. 21(1), 1–20 (2012)
  • Abo, H.: On three conjectures about the secant defectivity of classically studied varieties. In: Proceedings of the Algebraic Geometry Symposium...
  • Abo, H., Vannieuwenhoven, N.: Most secant varieties of tangential varieties to veronese varieties are nondefective. Trans. Am. Math. Soc....
  • Antonyan, L.V.: Classification of four-vectors of an eight-dimensional space. Trudy Sem. Vektor. Tenzor. Anal. 20, 144–161 (1981) MR622013
  • Arrondo, E., Bernardi, A., Marques, P.M., Mourrain, B.: Skew-symmetric tensor decomposition. Commun. Contemp. Math. 23(2), 1950061 (2021)
  • Baur, K., Draisma, J., de Graaf, W.A.: Secant dimensions of minimal orbits: computations and conjectures. Exp. Math. 16(2), 239–250 (2007)
  • Beelen, P., Ghorpade, S.R., Hoholdt, T.: Affine Grassmann codes. IEEE Trans. Inf. Theory 56(7), 3166–3176 (2010)
  • Bernardi, A., Vanzo, D.: A new class of non-identifiable skew-symmetric tensors. Annali di Matematica Pura ed Applicata (1923-) 197(5), 1499–1510...
  • Boralevi, A.: A note on secants of Grassmannians. Rend. Istit. Mat. Univ. Trieste 45, 67–72 (2013)
  • Casarotti, A., Mella, M.: Tangential weak defectiveness and generic identifiability. Int. Math. Res. Not. 2022(19), 15075–15091 (2022). arXiv:2009.00968
  • Catalisano, M.V., Geramita, A.V., Gimigliano, A.: On the rank of tensors, via secant varieties and fat points. In: Zero-Dimensional Schemes...
  • Catalisano, M.V., Geramita, A.V., Gimigliano, A.: Secant varieties of Grassmann varieties. Proc. Am. Math. Soc. 133(3), 633–642 (2005) (electronic)
  • Chiantini, L., Hauenstein, J.D., Ikenmeyer, C., Landsberg, J.M., Ottaviani, G.: Polynomials and the exponent of matrix multiplication. Bull....
  • Daleo, N.S., Hauenstein, J.D., Oeding, L.: Computations and equations for Segre-Grassmann hyper-surfaces. Port. Math. 73(1), 71–90 (2016)
  • Di Tullio, D., Gyawali, M.: A post-quantum signature scheme from the secant variety of the Grassmannian. In: Cryptology ePrint Archive (2020)
  • Fulton, W., Harris, J.: Representation Theory, A First Course, Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)
  • Geramita, A.V.: Inverse systems of fat points: Waring’s problem, secant varieties of Veronese varieties and parameter spaces for Gorenstein...
  • Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  • Hankerson, D., Hoffman, G., Leonard, D.A., Lindner, C.C., Phelps, K.T., Rodger, C.A., Wall, J.R.: Coding Theory and Cryptography: The Essentials....
  • Holweck, F., Oeding, L.: Hyperdeterminants from the E8 discriminant. J. Algebra 593, 622–650 (2022)
  • Holweck, F., Oeding, L.: Jordan decompositions of tensors (2022). arXiv:2206.13662
  • Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)
  • Landsberg, J.M.: Tensors: Geometry and Applications. Graduate Studies in Mathematics, vol. 128. American Mathematical Society, Providence...
  • Landsberg, J.M., Weyman, J.: On the ideals and singularities of secant varieties of Segre varieties. Bull. Lond. Math. Soc. 39(4), 685–697...
  • Massarenti, A., Rischter, R.: Non-secant defectivity via osculating projections. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 1–34 (2019)
  • McGillivray, B.: A probabilistic algorithm for the secant defect of Grassmann varieties. Linear Algebra Appl. 418(2–3), 708–718 (2006)
  • Nogin, D.Y.: Codes associated to Grassmannians. In: Arithmetic, Geometry and Coding Theory (Luminy, 1993), pp. 145–154 (2011)
  • Oeding, L.: A translation of “Classification of four-vectors of an 8-dimensional space,” by Antonyan, L.V., with an appendix by the translator,...
  • Ottaviani, G., Rubei, E.: Quivers and the cohomology of homogeneous vector bundles. Duke Math. J. 132(3), 459–508 (2006)
  • Postinghel, E.: A new proof of the Alexander-Hirschowitz interpolation theorem. Ann. Mat. Pura Appl. (4) 191(1), 77–94 (2012)
  • Ryan, C.: An application of Grassmannian varieties to coding theory. Congr. Numer. 57, 257–271 (1987)
  • Terracini, A.: Sulle Vk per cui la variet’a degli Sh (h + 1)-secanti ha dimensione minore dell’ordinario. Rend. Circ. Mat. Palermo Selecta...
  • Vinberg, È.B., Èlašvili, A.G.: A classification of the three-vectors of nine-dimensional space. Trudy Sem. Vektor. Tenzor. Anal. 18, 197–233...
  • Zak, F.L.: Tangents and Secants of Algebraic Varieties, Transl. Math. Monographs, vol. 127. Amer. Math. Soc., Providence (1993)