Skip to main content
SpringerLink
Log in

On the Multigraded Hilbert Function of Lines and Rational Curves in Multiprojective Spaces

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the multigraded Hilbert function of general configurations of lines in multiprojective spaces. In several cases we prove that this multigraded Hilbert function is the expected one. We make conjectures about other configurations and for small genus curves with a prescribed multidegree.

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. Aladpoosh, T.: Postulation of generic lines and one double line in \(\mathbb {P}_{n}\) in view of generic lines and one multiple linear space. Sel. Math. New Ser. 25, 9 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexander, J., Hirschowitz, A.: Interpolation on jets. J. Algebra 192, 412–417 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ballico, E.: Postulation of disjoint unions of lines and a few planes. J. Pure Appl. Algebra 215, 597–608 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ballico, E.: Postulation of disjoint unions of lines and double lines in \(\mathbb {P}^{3}\). Mediterr. J. Math. 9, 551–562 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biamonte, J.D., Bergholm, V.: Tensor networks in a nutshell. arXiv:1708.00006 (2017)

  6. Biamonte, J.D., Bergholm, V., Lanzagorta, M.: Tensor network methods for invariant theory. J. Phys. A: Math. Theor. 46, 475301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bocci, C., Chiantini, L., Ottaviani, G.: Refined methods for the identifiability of tensors. Ann. Mat. Pura Appl. 193(4), 1691–1702 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlini, E., Catalisano, M.V., Geramita, A.V.: Subspace arrangements, configurations of linear spaces and the quadrics containing them. J. Algebra 362, 70–83 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carlini, E., Catalisano, M.V., Geramita, A.V.: Bipolynomial Hilbert functions. J. Algebra 324, 758–781 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlini, E., Catalisano, M.V., Geramita, A.V.: 3-dimensional sundials. Centr. Eur. J. Math. 9, 949–971 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Matrix Anal. Appl. 35, 1265–1287 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On generic identifiability of symmetric tensors of subgeneric rank. Trans. Amer. Math. Soc. 369, 4021–4042 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chiantini, L., Ottavian, G., Vannieuwenhoven, N.: Effective criteria for specific identifiability of tensors and forms. SIAM J. Matrix Anal. Appl. 38, 656–681 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ciliberto, C., Miranda, R.: Interpolations on curvilinear schemes. J. Algebra 203, 677–678 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Critch, A., Morton, J.: Algebraic geometry of matrix product states. SIGMA 10, 095 (2014). arXiv:1210.2812(2012)

    MathSciNet  MATH  Google Scholar 

  16. Dumnicki, M., Harbourne, B., Szemberg, T., Tutaj-Gasińska, H.: Linear subspaces, symbolic powers and Nagata type conjectures. Adv. Math. 252, 471–491 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fatabbi, G., Harbourne, B., Lorenzini, A.: Inductively computable unions of fat linear subspaces. J. Pure Appl. Algebra 219, 5413–5425 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guardo, E., Harbourne, B., Van Tuyl, A.: Asymptotic resurgences for ideals of positive dimensional subschemes of projective space. Adv. Math. 246, 114–127 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hartshorne, R., Hirschowitz, A.: Droites en position générale dans l’espace projectif. In: Aroca, J.M., Buchweitz, R., Giusti, M., Merle, M (eds.) Algebraic Geometry, Proceedings at La Rábida 1981. Lecture Notes in Mathematics, vol. 961, pp 169–188. Springer, Berlin (1982)

  20. Hirschowitz, A.: Sur la postulation générique des courbes rationnelles. Acta Math. 146, 209–230 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kruskal, J.B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl. 18, 95–138 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  23. Larson, E.: The maximal rank conjecture for sections of curves. J. Algebra 555, 223–245 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Larson, E.: The maximal rank conjecture. arXiv:1711.04906(2017)

  25. Orús, R.: A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Ann. Phys. 349, 117–158 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Teitler, Z., Torrance, D.A.: Castelnuovo–Mumford regularity and arithmetic Cohen–Macaulayness of complete bipartite subspace arrangements. J. Pure Appl. Algebra 219, 2134–2138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author was partially supported by MIUR and GNSAGA of INdAM (Italy).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Trento, Trento, 38123, Povo (TN), Italy

    Edoardo Ballico

Authors
  1. Edoardo Ballico
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edoardo Ballico.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ballico, E. On the Multigraded Hilbert Function of Lines and Rational Curves in Multiprojective Spaces. Vietnam J. Math. 51, 435–449 (2023). https://doi.org/10.1007/s10013-021-00537-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-021-00537-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation