Advertisement

Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 109–113 | Cite as

Zero-dimensional complete intersections and their linear span in the Veronese embeddings of projective spaces

  • Edoardo BallicoEmail author
Open Access
Article
  • 52 Downloads

Abstract

Let \(\nu _{d,n}: \mathbb {P}^n\rightarrow \mathbb {P}^r\), \(r=\left( {\begin{array}{c}n+d\\ n\end{array}}\right) \), be the order d Veronese embedding. For any \(d_n\ge \cdots \ge d_1>0\) let \(\check{\eta }(n,d;d_1,\ldots ,d_n)\subseteq \mathbb {P}^r\) be the union of all linear spans of \(\nu _{d,n}(S)\) where \(S\subset \mathbb {P}^n\) is a finite set which is the complete intersection of hypersurfaces of degree \(d_1, \dots ,d_n\). For any \(q\in \check{\eta }(n,d;d_1,\ldots ,d_n)\), we prove the uniqueness of the set \(\nu _{d,n}(S)\) if \(d\ge d_1+\cdots +d_{n-1}+2d_n-n\) and q is not spanned by a proper subset of \(\nu _{d,n}(S)\). We compute \(\dim \check{\eta }(2,d;d_1,d_1)\) when \(d\ge 2d_1\).

Mathematics Subject Classification

14N05 15A69 

Notes

References

  1. 1.
    Ådlandsvik, B.: Joins and higher secant varieties. Math. Scand. 61, 213–222 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alexander, J.; Hirschowitz, A.: Polynomial interpolation in several variables. J. Algebra Geom. 4(2), 201–222 (1995)Google Scholar
  3. 3.
    Backelin, J.; Oneto, A.: On a class of power ideals. J. Pure Appl. Algebra 219(8), 3158–3180 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ballico, E.; Bernardi, A.: Symmetric tensor rank with a tangent vector: a generic uniqueness theorem. Proc. Am. Math. Soc. 140(10), 3377–3384 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ballico, E.; Bernardi, A.: Minimal decomposition of binary forms with respect to tangential projections. J. Algebra Appl. 12(6), 1350010, 8 pp. (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brambilla, M.C.; Ottaviani, G.: On the Alexander-Hirschowitz Theorem. J. Pure Appl. Algebra 212(5), 1229–1251 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buczyński, J.; Ginensky, A.; Landsberg, J.M.: Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture. J. Lond. Math. Soc. 88, 1–24 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Carlini, E.: Beyond Waring’s problem for forms: the binary decomposition. Rend. Sem. Mat. Univ. Pol. Torino 1 Polynom. Interp. Proceed. 63, 87–90 (2005)Google Scholar
  9. 9.
    Carlini, E.; Oneto, A.: Monomials as sums of \(k\)-th-powers of forms. Commun. Algebra 43(2), 650–658 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fröberg, R.; Ottaviani, G.; Shapiro, B.: On the Waring problem for polynomial rings. Proc. Natl. Acad. Sci. USA 109(15), 5600–5602 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)CrossRefGoogle Scholar
  12. 12.
    Iarrobino, A.; Kanev, V.: Power sums, Gorenstein algebras, and determinantal loci, vol. 1721. Lecture Notes in Mathematics. Springer, Berlin (1999) (Appendix C by Iarrobino and Steven L. Kleiman) Google Scholar
  13. 13.
    Landsberg, J.M.: Tensors: Geometry and Applications Graduate Studies in Mathematics. Am. Math. Soc. Providence, 1 28 (2012)Google Scholar
  14. 14.
    Libgober, A.S.; Woo, J.W.: Remarks on moduli spaces of complete intersections, The Lefschetz Centennial Conference, Part I: Proceedings on Algebraic Geometry, Contemporary Mathematics, 58, pp. 183–194. American Mathematical Society, Providence, RI (1986)Google Scholar
  15. 15.
    Lundqvist, S.; Oneto, A.; Reznik, B.; Shapiro, B.: On Generic and Maximal \(k\)-Ranks of Binary Forms. arXiv: 1711.05014.
  16. 16.
    Rathmann, J.: The uniform position principle for curves in characteristic \(p\). Math. Ann. 276(4), 565–579 (1987)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Scheiderer, C.: Sum of squares length of real forms. Math. Z. 286(1–2), 559–570 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TrentoPovoItaly

Personalised recommendations