Secant varieties of ℙ1 × ⋯ × ℙ1 (n-times) are not defective for n ≥ 5

Journal of Algebraic Geometry

Volumn 20, Issue 2, 2011, Pages 295-327

  Catalisano, Maria Virginia ^a   Geramita, Anthony V ^a,b   Gimigliano, Alessandro ^c  

^a UNIVERSITY OF GENOA   (Italy)

^b QUEEN S UNIVERSITY   (Canada)

^c UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79751483139     PISSN: 10563911     EISSN: None     Source Type: Journal    
DOI: 10.1090/S1056-3911-10-00537-0     Document Type: Article

References (27)

1. J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables. J. Algebraic Geom., 4(2):201-222, 1995. MR1311347 (96f:14065)
2. J. Alexander and A. Hirschowitz. An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math., 140(2):303-325, 2000. MR1756998 (2001i:14024)
3. H. Abo, G. Ottaviani, and C. Peterson. Induction for secant varieties for segre varieties. Available at http://front.math.ucdavis.edu/0607.5191, 2006.
4. Elizabeth S. Allman and John A. Rhodes. Molecular phylogenetics from an algebraic viewpoint. Statist. Sinica, 17(4):1299-1316, 2007. MR2398597 (2009e:62426)
5. 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. MR2388607 (2008m:60145) (Pubitemid 351199916)
6. 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. MR1440179 (99c:68002)
7. Luca Chiantini and Marc Coppens. Grassmannians of secant varieties. Forum Math., 13(5):615-628, 2001. MR1858491 (2002g:14079)
8. L. Chiantini and C. Ciliberto.Weakly defective varieties. Trans. Amer. Math. Soc., 354(1):151-178 (electronic), 2002. MR1859030 (2003b:14063)
9. 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. MR1930149 (2003g:14070)
10. 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. MR1930149 (2003g:14070)]
11. Linear Algebra Appl., 367:347-348, 2003. MR1976931
12. 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. MR2202248 (2007k:14109a)
13. 1. J. Pure Appl. Algebra, 201(1-3):367-380, 2005. MR2158764 (2006d:14060)
14. 1 and their secant varieties. Collect. Math., 58(1):1-24, 2007. MR2310544 (2008f:14069)
15. CoCoATeam. CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it, 2004.
16. Jan Draisma. A tropical approach to secant dimensions. J. Pure Appl. Algebra, 212(2):349-363, 2008. MR2357337 (2008j:14102)
17. Shmuel Friedland. On the generic rank of 3-tensors. Available at http://front.math.ucdavis.edu/0805.1959, 2008.
18. Dan Geiger, David Heckerman, Henry King, and Christopher Meek. Stratified exponential families: graphical models and model selection. Ann. Statist., 29(2):505-529, 2001. MR1863967 (2002h:60020)
19. Luis David Garcia, Michael Stillman, and Bernd Sturmfels. Algebraic geometry of Bayesian networks. J. Symbolic Comput., 39(3-4):331-355, 2005. MR2168286 (2006g:68242) (Pubitemid 40373617)
20. R. Hartshorne and A. Hirschowitz. Courbes rationnelles et droites en position générale. Ann. Inst. Fourier (Grenoble), 35(4):39-58, 1985. MR812318 (87e:14028)
21. Vassil Kanev. Chordal varieties of Veronese varieties and catalecticant matrices. J. Math. Sci. (New York), 94(1):1114-1125, 1999. Algebraic geometry, 9. MR1703911 (2001b:14078)
22. J. M. Landsberg. Geometry and the complexity of matrix multiplication. Bull. Amer. Math. Soc. (N.S.), 45(2):247-284, 2008. MR2383305 (2009b:68055)
23. J. M. Landsberg and L. Manivel. Generalizations of Strassen's equations for secant varieties of Segre varieties. Comm. Algebra, 36(2):405-422, 2008. MR2387532 (2009f:14109)
24. 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. MR2346950 (2008h:14055)
25. 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.
26. A. Terracini. Sulle vk per cui la varietà degli sh (h + 1)-seganti ha dimensione minore dell'ordinario. Rend. Circ. Mat. Palermo, 31:392-396, 1911.
27. 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. MR1234494 (94i:14053)