On the variety parameterizing completely decomposable polynomials

Thumbnail Image
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Elsevier Science
Google Scholar
Research Projects
Organizational Units
Journal Issue
The purpose of this paper is to relate the variety parameterizing completely decomposable homogeneous polynomials of degree d in n + 1 variables on an algebraically closed field, called Split(d)(P(n)), with the Grassmannian of (n - 1)-dimensional projective subspaces of P(n+d-1). We compute the dimension of some secant varieties to Split(d)(P(n)). Moreover by using an invariant embedding of the Veronese variety into the Plucker space, we are able to compute the intersection of G(n - 1, n + d - 1) with Split(d)(P(n)), some of its secant varieties, the tangential variety and the second osculating space to the Veronese variety.
[1] H. Abo, G. Ottaviani, C. Peterson, Induction for secant varieties of Segre varieties, Transactions of the American Mathematical Society 361 (2009) 767–792. [2] H. Abo, G. Ottaviani, C. Peterson, Non-Defectivity of Grassmannians of planes, Journal of Algebraic Geometry (in press). Preprint: [3] B. Ådlandsvik, Varieties with an extremal number of degenerate higher secant varieties, Journal für die Reine und Angewandte Mathematik 392 (1988) 16–26. [4] J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, Journal of Algebraic Geometry 4 (1995) 201–222. [5] E. Arrondo, R. Paoletti, Characterization of Veronese varieties via projections in Grassmannians, in: C. Ciliberto, A.V. Geramita, R. Miró-Roig, Ranestad, W. De Gruyter (Eds.), Projective Varieties with Unexpected Properties, 2005, pp. 1–12. [6] A. Bernardi, Varieties parameterizing forms and their secant varieties, Ph.D. Thesis 2006, Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, Bollettino UMI, 2007. [7] A. Bernardi, M.V. Catalisano, A. Gimigliano, M. Idà, Osculating varieties of veronese varieties and their higher secant varieties, Canadian Journal of Mathematics 59 (2007) 488–502. [8] M.C Brambilla, G. Ottaviani, On the Alexander–Hirschowitz theorem, Journal of Pure and Applied Algebra 212 (2008) 1229–1251. [9] A. Capani, G. Niesi, L. Robbiano, CoCoA, A system for doing Computations in Commutative Algebra. Available via anonymous ftp from: [10] E. Carlini, Codimension one decompositions and Chow varieties, in: C. Ciliberto, B. Harbourne, R. Miró-Roig, K. Ranestad, A. Geramita, W. De Gruyter (Eds.), Projective Varieties with Unexpected Properties, 2005, pp. 67–79. [11] M.V. Catalisano, A.V. Geramita, A. Gimigliano, Secant varieties of Grassmann varieties, in: Proceedings of the American Mathematical Society, vol. 133, 2004, pp. 633–642. [12] M.V. Catalisano, A.V. Geramita, A. Gimigliano, Higher secant varieties of Segre–Veronese varieties, in: Atti del Convegno: Varieties with Unexpected Properties, Siena, Giugno 2004, W. de Gruyter, Berlin, 2005, pp. 81–107. [13] M.V. Catalisano, A.V. Geramita, A. Gimigliano, Tensor rank, secant varieties to Segre varieties and fat points in multiprojective spaces, Linear Algebra and its Applications 355 (2002) 263–285; see also the errata of the publisher: 367 (2003) 347–348. [14] R. Ehrenborg, On apolarity and generic canonical forms, Journal of Algebra 213 (1999) 167–194. [15] I.M. Gel’fand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, in: Mathematics: Theory & Applications, Birkhóauser Boston Inc., Boston, MA, 1994. [16] B. Mc Gillivray, A probabilistic algorithm for the secant defect of Grassmann varieties, Linear Algebra and its Applications 418 (2006) 708–718. [17] F.D. Murnaghan, The Theory of Group Representations, The Johns Hopkins Press, Baltimore, 1938. [18] A. Terracini, Sulle Vk per cui la varietà degli Sh(h + 1)-seganti ha dimensione minore dell’ordinario, Rendiconti del Circolo Matematico di Palermo 31 (1911) 392–396.