On the identifiability of binary Segre products
Authors:
Cristiano Bocci and Luca Chiantini
Journal:
J. Algebraic Geom. 22 (2013), 1-11
DOI:
https://doi.org/10.1090/S1056-3911-2011-00592-4
Published electronically:
November 22, 2011
MathSciNet review:
2993044
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Abstract |
References |
Additional Information
Abstract: We prove that a product of $m>5$ copies of $\mathbb {P}^1$, embedded in the projective space $\mathbb {P}^r$ by the standard Segre embedding, is $k$-identifiable (i.e. a general point of the secant variety $S^k(X)$ is contained in only one $(k+1)$-secant $k$-space), for all $k$ such that $k+1\leq 2^{m-1}/m$.
References
- Hirotachi Abo, Giorgio Ottaviani, and Chris Peterson, Induction for secant varieties of Segre varieties, Trans. Amer. Math. Soc. 361 (2009), no. 2, 767–792. MR 2452824, DOI https://doi.org/10.1090/S0002-9947-08-04725-9
- Bjørn Ådlandsvik, Joins and higher secant varieties, Math. Scand. 61 (1987), no. 2, 213–222. MR 947474, DOI https://doi.org/10.7146/math.scand.a-12200
- Elizabeth S. Allman, Catherine Matias, and John A. Rhodes, Identifiability of parameters in latent structure models with many observed variables, Ann. Statist. 37 (2009), no. 6A, 3099–3132. MR 2549554, DOI https://doi.org/10.1214/09-AOS689
- C. Bocci, L. Chiantini, Appendix to “On the identifiability of binary Segre products”, Available at: www.mat.unisi.it/personalpages/chiantini/p1app.pdf.
- M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263–285. MR 1930149, DOI https://doi.org/10.1016/S0024-3795%2802%2900352-X
- Maria Virginia Catalisano, Anthony V. Geramita, and Alessandro Gimigliano, Secant varieties of $\Bbb P^1\times \dots \times \Bbb P^1$ ($n$-times) are not defective for $n\geq 5$, J. Algebraic Geom. 20 (2011), no. 2, 295–327. MR 2762993, DOI https://doi.org/10.1090/S1056-3911-10-00537-0
- L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), no. 1, 151–178. MR 1859030, DOI https://doi.org/10.1090/S0002-9947-01-02810-0
- Luca Chiantini and Ciro Ciliberto, On the classification of defective threefolds, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 131–176. MR 2202251
- Luca Chiantini and Ciro Ciliberto, On the concept of $k$-secant order of a variety, J. London Math. Soc. (2) 73 (2006), no. 2, 436–454. MR 2225496, DOI https://doi.org/10.1112/S0024610706022630
- Ciro Ciliberto, Massimiliano Mella, and Francesco Russo, Varieties with one apparent double point, J. Algebraic Geom. 13 (2004), no. 3, 475–512. MR 2047678, DOI https://doi.org/10.1090/S1056-3911-03-00355-2
- M. Dale, Terracini’s lemma and the secant variety of a curve, Proc. London Math. Soc. (3) 49 (1984), no. 2, 329–339. MR 748993, DOI https://doi.org/10.1112/plms/s3-49.2.329
- Ryan Elmore, Peter Hall, and Amnon Neeman, An application of classical invariant theory to identifiability in nonparametric mixtures, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 1–28 (English, with English and French summaries). MR 2141286
- D. Grayson, M. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at: www.math.uiuc.edu/Macaulay2/.
- Joseph B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics, Linear Algebra Appl. 18 (1977), no. 2, 95–138. MR 444690, DOI https://doi.org/10.1016/0024-3795%2877%2990069-6
- 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 (1911), 392–396.
- F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494
References
- H. Abo, G. Ottaviani, C. Peterson, Induction for secant varieties of Segre varieties, Trans. Amer. Math. Soc. 361 (2009), 767–792. MR 2452824 (2010a:14088)
- B. Adlansdvik, Joins and Higher secant varieties, Math. Scand. 61 (1987), 213–222. MR 947474 (89j:14030)
- E.S. Allman, C. Matias, J.A. Rhodes, Identifiability of parameters in latent structure models with many observed variables. Ann. Statist. 37 (2009), 3099–3132. MR 2549554 (2010m:62029)
- C. Bocci, L. Chiantini, Appendix to “On the identifiability of binary Segre products”, Available at: www.mat.unisi.it/personalpages/chiantini/p1app.pdf.
- M. V. Catalisano, A. V. Geramita, A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263–285. MR 1930149 (2003g:14070)
- M. V. Catalisano, A. V. Geramita, A. Gimigliano, Secant Varieties of $(\mathbb {P} ^1)\times \dots \times (\mathbb {P} ^1)$ ($n$-times) are NOT Defective for $n \geq 5$, J. Algegraic Geom. 20 (2011), 295–327. MR 2762993
- L. Chiantini, C. Ciliberto, Weakly defective varieties, Trans. Amer. Math. Soc. 354 (2002), 151–178. MR 1859030 (2003b:14063)
- L. Chiantini, C. Ciliberto, On the classification of defective threefolds, in Projective Varieties with Unexpected Properties, Proceedings of the Siena Conference, C. Ciliberto, A. V. Geramita, B. Harbourne, R. M. Mirò–Roig, K. Ranestad, ed., W. de Gruyter (2005), 131–176. MR 2202251 (2006k:14068)
- L. Chiantini, C. Ciliberto, On the concept of $k$-secant order of a variety, J. London Math. Soc. 73 (2006), 436–454. MR 2225496 (2007k:14110)
- C. Ciliberto, M. Mella, F. Russo, Varieties with one apparent double point, J. Algebraic Geom. 13 (2004), 475–512. MR 2047678 (2005b:14078)
- M. Dale, Terracini’s lemma and the secant variety of a curve, Proc. London Math. Soc. 49 (1984), 329–339. MR 748993 (85g:14066)
- R. Elmore, P. Hall, A. Neeman, An application of classical invariant theory to identifiability in non-parametric mixtures, Ann. Inst. Fourier 55 (2005), 1–28. MR 2141286 (2006m:62056)
- D. Grayson, M. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at: www.math.uiuc.edu/Macaulay2/.
- J. B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics, Lin. Alg. Applic. 18 (1977), 95–138. MR 0444690 (56:3040)
- 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 (1911), 392–396.
- F. L. Zak, Tangents and secants of varieties, AMS Bookstore publications, Transl. Math. Monog. 127 (1993). MR 1234494 (94i:14053)
Additional Information
Cristiano Bocci
Affiliation:
Universitá degli Studi di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, I – 53100 Siena, Italy
Email:
cristiano.bocci@unisi.it
Luca Chiantini
Affiliation:
Universitá degli Studi di Siena, Dipartimento di Scienze Matematiche e Informatiche, Pian dei Mantellini, 44, I – 53100 Siena, Italy
MR Author ID:
194958
ORCID:
0000-0001-5776-1335
Email:
chiantini@unisi.it
Received by editor(s):
December 30, 2009
Received by editor(s) in revised form:
February 21, 2011, and March 9, 2011
Published electronically:
November 22, 2011
