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
DOI:
https://doi.org/10.1090/S1056-3911-10-00537-0
Published electronically:
March 25, 2010
MathSciNet review:
2762993
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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
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References
- J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables. J. Algebraic Geom., 4(2):201–222, 1995. MR 1311347 (96f:14065)
- 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)
- H. Abo, G. Ottaviani, and C. Peterson. Induction for secant varieties for segre varieties. Available at http://front.math.ucdavis.edu/0607.5191, 2006.
- Elizabeth S. Allman and John A. Rhodes. Molecular phylogenetics from an algebraic viewpoint. Statist. Sinica, 17(4):1299–1316, 2007. MR 2398597 (2009e:62426)
- 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)
- 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)
- Luca Chiantini and Marc Coppens. Grassmannians of secant varieties. Forum Math., 13(5):615–628, 2001. MR 1858491 (2002g:14079)
- L. Chiantini and C. Ciliberto. Weakly defective varieties. Trans. Amer. Math. Soc., 354(1):151–178 (electronic), 2002. MR 1859030 (2003b:14063)
- 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)
- 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
- 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)
- M. V. Catalisano, A. V. Geramita, and A. Gimigliano. Higher secant varieties of the Segre varieties $\mathbb P^ 1\times \dots \times \mathbb P^ 1$. J. Pure Appl. Algebra, 201(1-3):367–380, 2005. MR 2158764 (2006d:14060)
- M. V. Catalisano, A. V. Geramita, and A. Gimigliano. Segre-Veronese embeddings of $\mathbb P^ 1\times \mathbb P^ 1\times \mathbb P^ 1$ and their secant varieties. Collect. Math., 58(1):1–24, 2007. MR 2310544 (2008f:14069)
- CoCoATeam. CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it, 2004.
- Jan Draisma. A tropical approach to secant dimensions. J. Pure Appl. Algebra, 212(2):349–363, 2008. MR 2357337 (2008j:14102)
- Shmuel Friedland. On the generic rank of 3-tensors. Available at http://front. math.ucdavis.edu/0805.1959, 2008.
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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)
- 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.
- 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.
- 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
Email:
catalisano@diptem.unige.it
Anthony V. Geramita
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, and Dipartimento di Matematica, Università di Genova,Genoa, Italy
MR Author ID:
72575
Email:
anthony.geramita@gmail.com
Alessandro Gimigliano
Affiliation:
Dipartimento di Matematica and CIRAM, Università di Bologna, 40126 Bologna, Italy
Email:
gimiglia@dm.unibo.it
Received by editor(s):
September 27, 2008
Received by editor(s) in revised form:
March 12, 2009
Published electronically:
March 25, 2010
