Skip to main content
SpringerLink
Log in

Identifiability and singular locus of secant varieties to Grassmannians

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

Secant varieties are among the main protagonists in tensor decomposition, whose study involves both pure and applied mathematic areas. Grassmannians are the building blocks for skewsymmetric tensors. Although they are ubiquitous in the literature, the geometry of their secant varieties is not completely understood. In this work we determine the singular locus of the secant variety of lines to a Grassmannian Gr(k, V) using its structure as \({{\,\textrm{SL}\,}}(V)\)-variety. We solve the problems of identifiability and tangential-identifiability of points in the secant variety: as a consequence, we also determine the second Terracini locus to a Grassmannian.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abo, H., Ottaviani, G., Peterson, C.: Non-defectivity of Grassmannians of planes. J. Algebraic Geom. 21(1), 1–20 (2009)

    Article  MathSciNet  Google Scholar 

  2. Arrondo, E., Bernardi, A., Marques, P.M., Mourrain, B., Skew-symmetric tensor decomposition. In: Communications in Contemporary Mathematics 23.02 (2021), DOI: https://doi.org/10.1142/S0219199719500615

  3. Ballico, E., Bernardi, A., Santarsiero, P.: Terracini locus for three points on a Segre variety. arXiv preprint arXiv:2012.00574 (2020)

  4. Ballico, E., Chiantini, L.: On the terracini locus of projective varieties. Milan J. Math. 89(1), 1–17 (2021)

    Article  MathSciNet  Google Scholar 

  5. Baur, K., Draisma, J., De Graaf, W.: Secant dimensions of minimal orbits: computations and conjectures. Exp. Math. 16(2), 239–250 (2007)

    Article  MathSciNet  Google Scholar 

  6. Bernardi, A., Carlini, E., Catalisano, M.V., Gimigliano, A., Oneto, A.: The hitchhiker guide to: Secant varieties and tensor decomposition. Mathematics 6(12), 314 (2018)

    Article  Google Scholar 

  7. Bernardi, A., Carusotto, I.: Algebraic geometry tools for the study of entanglement: an application to spin squeezed states. J. Phys. A Math. Theoret. 45(10), 105304 (2012)

    Article  MathSciNet  Google Scholar 

  8. Bernardi, A., Vanzo, D.: A new class of non-identifiable skew-symmetric tensors. In: Annali di Matematica Pura ed Applicata (1923-) 197(5), pp. 1499–1510 (2018)

  9. Bidleman, D., Oeding, L.: Restricted Secant Varieties of Grassmannians. In: Collectanea Mathematica (2023)

  10. Boralevi, A.: A note on secants of Grassmannians. Rend. Istit. Mat. Univ. Trieste 45, 67–72 (2013)

    MathSciNet  Google Scholar 

  11. Catalisano, M., Geramita, A., Gimigliano, A.: Secant varieties of Grassmann varieties. Proc. Am. Math. Soc. 133(3), 633–642 (2005)

    Article  MathSciNet  Google Scholar 

  12. Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. 12(5), 1267–1291 (2010)

    Article  MathSciNet  Google Scholar 

  13. Comon, P.: Tensors: a brief introduction. IEEE Signal Process. Mag. 31(3), 44–53 (2014)

    Article  Google Scholar 

  14. Fulton, W., Harris, J.: Representation Theory: A First Course. Vol. 129. Springer Science & Business Media (2013)

  15. Furukawa, K., Han, K.: On the singular loci of higher secant varieties of Veronese embeddings. preprint arXiv:2111.03254 (2021)

  16. Galgano, V.: Identifiability and singular locus of secant varieties to spinor varieties. In: preprint arXiv:2302.05295 (2023)

  17. Galuppi, F., Oneto, A.: Secant non-defectivity via collisions of fat points. Adv. Math. 409, 108657 (2022)

    Article  MathSciNet  Google Scholar 

  18. Han, K.: On singularities of third secant varieties of Veronese embeddings. Linear Algebra Appl. 544, 391–406 (2018)

    Article  MathSciNet  Google Scholar 

  19. Harris, J.: Algebraic Geometry: A First Course. Vol. 133. Springer Science & Business Media (2013)

  20. Kanev, V.: Chordal varieties of Veronese varieties and catalecticant matrices. J. Math. Sci. 94(1), 1114–1125 (1999)

    Article  MathSciNet  Google Scholar 

  21. Laface, A., Postinghel, E.: Secant Varieties of Segre-Veronese embeddings of (P1)r. Mathematische Annalen 356(4), 1455–1470 (2013)

    Article  MathSciNet  Google Scholar 

  22. Landsberg, J.M.: Tensors: Geometry and Applications, vol. 128. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2012)

    Google Scholar 

  23. Landsberg, J.M., Manivel, L.: Legendrian varieties. Asian J. Math. 11(3), 341–360 (2007)

    Article  MathSciNet  Google Scholar 

  24. Lim, L.-H.: Tensors in computations. Acta Numer. 30, 555–764 (2021)

    Article  MathSciNet  Google Scholar 

  25. Manivel, L., Michalek, M.: Secants of minuscule and cominuscule minimal orbits. Linear Algebra Appl. 481, 288–312 (2015)

    Article  MathSciNet  Google Scholar 

  26. Michalek, M., Oeding, L., Zwiernik, P.: Secant cumulants and toric geometry. Int. Math. Res. Not. 2015(12), 4019–4063 (2015)

    MathSciNet  Google Scholar 

  27. Ottaviani, G., Reichenbach, P.: Tensor rank and complexity. arXiv preprint, arXiv:2004.01492 (2020)

  28. Russo, F.: On the Geometry of Some Special Projective Varieties. Springer International Publishing (2016)

  29. Ullery, B.: On the normality of secant varieties. Adv. Math. 288, 631–647 (2016)

    Article  MathSciNet  Google Scholar 

  30. Vermeire, P.: Some results on secant varieties leading to a geometric flip construction. Compos. Math. 125(3), 263–282 (2001)

    Article  MathSciNet  Google Scholar 

  31. Zak, F.L.: Tangents and secants of algebraic varieties. Vol. 127. American Mathematical Soc. (1993)

Download references

Acknowledgements

The authors warmly thank Alessandra Bernardi and Giorgio Ottaviani for the several and helpful discussions. Sincere thanks are due also to Laurent Manivel and Mateusz Michałek for their useful suggestions, as well as to the anonymous referees for the interesting and detailed remarks which allowed us to improve our work. The authors are members of the Italian GNSAGA-INdAM.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38123, Trento, Italy

    Vincenzo Galgano & Reynaldo Staffolani

Authors
  1. Vincenzo Galgano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reynaldo Staffolani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincenzo Galgano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galgano, V., Staffolani, R. Identifiability and singular locus of secant varieties to Grassmannians. Collect. Math. (2024). https://doi.org/10.1007/s13348-023-00429-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13348-023-00429-1

Keywords

Mathematics Subject Classification

Search

Navigation