Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 109–113

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

• Edoardo Ballico
Open Access
Article

## 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 $$d\ge 2d_1$$.

14N05 15A69

## References

1. 1.
Ådlandsvik, B.: Joins and higher secant varieties. Math. Scand. 61, 213–222 (1987)
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)
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)
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)
6. 6.
Brambilla, M.C.; Ottaviani, G.: On the Alexander-Hirschowitz Theorem. J. Pure Appl. Algebra 212(5), 1229–1251 (2008)
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)
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)
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)
11. 11.
Hartshorne, R.: Algebraic Geometry. Springer, Berlin (1977)
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)
17. 17.
Scheiderer, C.: Sum of squares length of real forms. Math. Z. 286(1–2), 559–570 (2017)