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About the Defectivity of Certain Segre–Veronese Varieties

Published online by Cambridge University Press:  20 November 2018

Silvia Abrescia
Silvia Abrescia*
Affiliation:
Dipartimento di Matematica, Università di Bologna, 40137 Bologna, Italy e-mail:abrescia@dm.unibo.it
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Abstract

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We study the regularity of the higher secant varieties of ${{\mathbb{P}}^{1}}\times {{\mathbb{P}}^{n}}$, embedded with divisors of type $\text{(}d\text{,}\,\text{2)}$ and $(d,3)$. We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of ${{\mathbb{P}}^{n}}$ is not Grassmann defective.

Keywords

14N1514N0514M12Waring problemSegre–Veronese embeddingsecant varietyGrassmann defectivity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 5 , October 2008 , pp. 961 - 974
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Ådlandsvik, B., Varieties with an extremal number of degenerate higher secant varieties. J. Reine Angew. Math. 392(1988), 16–26.Google Scholar
[2] Alexander, J. and Hirschowitz, A., Polynomial interpolation in several variables. J. Algebraic Geom. 4(1995), no. 2, 201–222.Google Scholar
[3] Alexander, J. and Hirschowitz, A., An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math. 140(2000), no. 2, 303–325.Google Scholar
[4] Bocci, C., Special effect varieties in higher dimension. Collect. Math. 56(2005) no. 3, 299–326.Google Scholar
[5] Carlini, E. and Catalisano, M. V., Existence results for rational normal curves. J. Lond. Math. Soc. (2) 76(2007), no. 1, 73–86.Google Scholar
[6] Catalisano, M. V., Geramita, A. V., and Gimigliano, A., Higher secant varieties of Segre-Veronese varieties. In: Projective Varieties with Unexpected Properties,Walter de Gruyter, Berlin, 2005, pp. 81–107.Google Scholar
[7] Catalisano, M. V., Geramita, A. V., and Gimigliano, A., Higher secant varieties of the Segre varieties ℙ1 × · · · × ℙ1. J. Pure Appl. Algebra 201(2005), no. 1-3, 367–380.Google Scholar
[8] Chiantini, L., Lectures on the structures of projective embeddings. Rend. Sem. Mat. Univ. Politec. Torino 62(2004), no. 4, 335–388.Google Scholar
[9] Chiantini, L., and Ciliberto, C., The classification of (1, k)-defective surfaces. Geom. Dedicata 111(2005), 107–123.Google Scholar
[10] Chiantini, L., and Ciliberto, C., Weakly defective varieties. Trans. Amer. Math. Soc. 354(2001), no. 1, 151–178.Google Scholar
[11] Chiantini, L., and Coppens, M., Grassmannians of secant varieties. ForumMath. 13(2001), no. 5, 615–628.Google Scholar
[12] Ciliberto, C. and Miranda, R., The Segre and Harbourne-Hirschowitz Conjectures. In: Applications of Algebraic Geometry to Coding Theory, Physics and Computation. NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer, Dordrecht, 2001, 37–51.Google Scholar
[13] CoCoATeam, CoCoA: A System for Doing Computations in Commutative Algebra. http://cocoa.dima.unige.itGoogle Scholar
[14] Dionisi, C. and Fontanari, C., Grassman defectivity à la Terracini, Matematiche, 56(2001), no. 2, 245–255.Google Scholar
[15] Fontanari, C., On Waring's problem for partially symmetric tensors. Variations on a theme of Mella. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52(2006), no. 1, 37–43.Google Scholar
[16] Harris, J., A bound on the geometric genus of projective varieties. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8(1981), no. 1, 35–68.Google Scholar
[17] Hirschowitz, A., La méthode d’Horace pour l’interpolation à plusieurs variables. Manuscripta Math. 50(1985), 337–388.Google Scholar
[18] Hirschowitz, A., Maple.http://www.maplesoft.com.Google Scholar
[19] Mella, M., Singularities of linear systems and the Waring problem. Trans. Amer. Math. Soc. 358(2006), no. 12, 5523–5538.Google Scholar