Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Secant varieties of $ {\mathbb{P}^1}\times \cdots \times {\mathbb{P}^1}$ ($ n$-times) are NOT defective for $ n \geq 5$

Authors: Maria Virginia Catalisano, Anthony V. Geramita and Alessandro Gimigliano
Journal: J. Algebraic Geom. 20 (2011), 295-327
Published electronically: March 25, 2010
MathSciNet review: 2762993
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $ V_n$ be the Segre embedding of $ {\mathbb{P}^1}\times \cdots \times {\mathbb{P}^1}$ ($ n$ times). We prove that the higher secant varieties $ \sigma_s(V_n)$ always have the expected dimension, except for $ \sigma_3(V_4)$, which is of dimension 1 less than expected.

References [Enhancements On Off] (What's this?)

  • [AH95] J. Alexander and A. Hirschowitz.
    Polynomial interpolation in several variables.
    J. Algebraic Geom., 4(2):201-222, 1995. MR 1311347 (96f:14065)
  • [AH00] J. Alexander and A. Hirschowitz.
    An asymptotic vanishing theorem for generic unions of multiple points.
    Invent. Math., 140(2):303-325, 2000. MR 1756998 (2001i:14024)
  • [AOP06] H. Abo, G. Ottaviani, and C. Peterson.
    Induction for secant varieties for segre varieties.
    Available at, 2006.
  • [AR07] Elizabeth S. Allman and John A. Rhodes.
    Molecular phylogenetics from an algebraic viewpoint.
    Statist. Sinica, 17(4):1299-1316, 2007. MR 2398597 (2009e:62426)
  • [AR08] Elizabeth S. Allman and John A. Rhodes.
    Phylogenetic ideals and varieties for the general Markov model.
    Adv. in Appl. Math., 40(2):127-148, 2008. MR 2388607 (2008m:60145)
  • [BCS97] Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi.
    Algebraic complexity theory, volume 315 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
    Springer-Verlag, Berlin, 1997.
    With the collaboration of Thomas Lickteig. MR 1440179 (99c:68002)
  • [CC01] Luca Chiantini and Marc Coppens.
    Grassmannians of secant varieties.
    Forum Math., 13(5):615-628, 2001. MR 1858491 (2002g:14079)
  • [CC02] L. Chiantini and C. Ciliberto.
    Weakly defective varieties.
    Trans. Amer. Math. Soc., 354(1):151-178 (electronic), 2002. MR 1859030 (2003b:14063)
  • [CGG02] M. V. Catalisano, A. V. Geramita, and A. Gimigliano.
    Ranks of tensors, secant varieties of Segre varieties and fat points.
    Linear Algebra Appl., 355:263-285, 2002. MR 1930149 (2003g:14070)
  • [CGG03] M. V. Catalisano, A. V. Geramita, and A. Gimigliano.
    Publisher's erratum to: ``Ranks of tensors, secant varieties of Segre varieties and fat points'' [Linear Algebra Appl. 355 (2002), 263-285. MR 1930149 (2003g:14070)]
    Linear Algebra Appl., 367:347-348, 2003.MR 1976931
  • [CGG05a] M. V. Catalisano, A. V. Geramita, and A. Gimigliano.
    Higher secant varieties of Segre-Veronese varieties.
    In Projective varieties with unexpected properties, pages 81-107. Walter de Gruyter GmbH & Co. KG, Berlin, 2005. MR 2202248 (2007k:14109a)
  • [CGG05b] M. V. Catalisano, A. V. Geramita, and A. Gimigliano.
    Higher secant varieties of the Segre varieties $ \mathbb{P}\sp 1\times\dots\times\mathbb{P}\sp 1$.
    J. Pure Appl. Algebra, 201(1-3):367-380, 2005. MR 2158764 (2006d:14060)
  • [CGG07] M. V. Catalisano, A. V. Geramita, and A. Gimigliano.
    Segre-Veronese embeddings of $ \mathbb{P}\sp 1\times\mathbb{P}\sp 1\times\mathbb{P}\sp 1$ and their secant varieties.
    Collect. Math., 58(1):1-24, 2007. MR 2310544 (2008f:14069)
  • [CoC04] CoCoATeam.
    CoCoA: a system for doing Computations in Commutative Algebra.
    Available at, 2004.
  • [Dra08] Jan Draisma.
    A tropical approach to secant dimensions.
    J. Pure Appl. Algebra, 212(2):349-363, 2008. MR 2357337 (2008j:14102)
  • [Fri08] Shmuel Friedland.
    On the generic rank of 3-tensors. Available at http://front., 2008.
  • [GHKM01] Dan Geiger, David Heckerman, Henry King, and Christopher Meek.
    Stratified exponential families: graphical models and model selection.
    Ann. Statist., 29(2):505-529, 2001. MR 1863967 (2002h:60020)
  • [GSS05] Luis David Garcia, Michael Stillman, and Bernd Sturmfels.
    Algebraic geometry of Bayesian networks.
    J. Symbolic Comput., 39(3-4):331-355, 2005. MR 2168286 (2006g:68242)
  • [HH85] R. Hartshorne and A. Hirschowitz.
    Courbes rationnelles et droites en position générale.
    Ann. Inst. Fourier (Grenoble), 35(4):39-58, 1985. MR 812318 (87e:14028)
  • [Kan99] Vassil Kanev.
    Chordal varieties of Veronese varieties and catalecticant matrices.
    J. Math. Sci. (New York), 94(1):1114-1125, 1999.
    Algebraic geometry, 9. MR 1703911 (2001b:14078)
  • [Lan08] J. M. Landsberg.
    Geometry and the complexity of matrix multiplication.
    Bull. Amer. Math. Soc. (N.S.), 45(2):247-284, 2008. MR 2383305 (2009b:68055)
  • [LM08] J. M. Landsberg and L. Manivel.
    Generalizations of Strassen's equations for secant varieties of Segre varieties.
    Comm. Algebra, 36(2):405-422, 2008. MR 2387532 (2009f:14109)
  • [LW07] J. M. Landsberg and Jerzy Weyman.
    On the ideals and singularities of secant varieties of Segre varieties.
    Bull. Lond. Math. Soc., 39(4):685-697, 2007. MR 2346950 (2008h:14055)
  • [Pal09] F. Palatini.
    Sulle varietà algebriche per le quali sono di dimensione minore dell' ordinario, senza riempire lo spazio ambiente, una o alcuna delle varietà formate da spazi seganti.
    Atti Accad. Torino Cl. Scienze Mat. Fis. Nat., 44:362-375, 1909.
  • [Ter11] A. Terracini.
    Sulle $ v_k$ per cui la varietà degli $ s_h$ $ (h+1)$-seganti ha dimensione minore dell'ordinario.
    Rend. Circ. Mat. Palermo, 31:392-396, 1911.
  • [Zak93] F. L. Zak.
    Tangents and secants of algebraic varieties, volume 127 of Translations of Mathematical Monographs.
    American Mathematical Society, Providence, RI, 1993.
    Translated from the Russian manuscript by the author. MR 1234494 (94i:14053)

Additional Information

Maria Virginia Catalisano
Affiliation: DIPTEM - Dipartimento di Ingegneria della Produzione, Termoenergetica e Modelli Matematici, Piazzale Kennedy, pad. D 16129 Genoa, Italy

Anthony V. Geramita
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, and Dipartimento di Matematica, Università di Genova,Genoa, Italy

Alessandro Gimigliano
Affiliation: Dipartimento di Matematica and CIRAM, Università di Bologna, 40126 Bologna, Italy

Received by editor(s): September 27, 2008
Received by editor(s) in revised form: March 12, 2009
Published electronically: March 25, 2010

American Mathematical Society