Segre-Veronese embeddings of P1 × P1 × P1 and their secant varieties

• Autores: Maria Virginia Catalisano, Anthony V. Geramita, A. Gimigliano
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 58, Fasc. 1, 2007, págs. 1-24
• Idioma: inglés
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• Resumen

  • In this paper we compute the dimension of all the sth higher secant varieties of the SegreVeronese embeddings Yd of the product P1 × P1 × P1 in the projective space PN via divisors of multidegree d = (a, b, c) (N = (a + 1)(b+1)(c+1)-1). We find that Yd has no deficient higher secant varieties, unless d = (2, 2, 2) and s = 7, or d = (2h, 1, 1) and s = 2h + 1, with defect 1 in both cases

• Referencias bibliográficas
  • J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), 303–325.
  • P. Bürgisser, M. Clausen, and M.A. Shokrollahi, Algebraic Complexity Theory, Springer-Verlag, Berlin, 1997.
  • A. Capani, G. Niesi, and L. Robbiano, CoCoA, a system for doing computations in commutative algebra (available via anonymous ftp from: cocoa.dima.unige.it).
  • M.V. Catalisano, A.V. Geramita, and A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl....
  • M.V. Catalisano, A.V. Geramita, and A. Gimigliano, Higher secant varieties of Segre-Veronese varieties, Projective Varieties with Unexpected...
  • M.V. Catalisano, A.V. Geramita, and A. Gimigliano, Higher secant varieties of the Segre varieties $\mathbb{P}^1 \times \cdots \times \mathbb{P}^1$,...
  • K. Chandler, A brief proof of a maximal rank theorem for generic double points in projective space, Trans. Amer. Math. Soc. 353 (2001), 1907–1920.
  • L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), 151–178.
  • L. Chiantini and C. Ciliberto, On the classification of defective threefolds, Projective Varieties with Unexpected Properties, Walter de Gruyter...
  • L. Chiantini and C. Ciliberto, The classification of $(1,k)$-defective surfaces, Geom. Dedicata 111 (2005), 107–123.
  • G. Fatabbi and A. Lorenzini, On a sharp bound for the regularity index of any set of fat points, J. Pure Appl. Algebra 161 (2001), 91–111.
  • L.D. Garcia, H. Stillman, and B. Sturmfels, Algebraic geometry of Bayesian networks, (preprint, 2003).
  • D. Geiger, D. Heckerman, H. King, and C. Meek, Stratified exponential families: graphical models and model selection, Ann. Statist. 29 (2001),...
  • A. Terracini, Sulle $V\_k$ per cui la varietà degli $S\_h$ $(h+1)$-seganti ha dimensione minore dell’ordinario, Rend. Circ. Mat. Palermo...
  • A. Terracini, Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari, Ann. Mat. Pura Appl. 24 (1915),...