Most secant varieties of tangential varieties to Veronese varieties are nondefective
HTML articles powered by AMS MathViewer
- by Hirotachi Abo and Nick Vannieuwenhoven PDF
- Trans. Amer. Math. Soc. 370 (2018), 393-420 Request permission
Abstract:
We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the $d$th Veronese embedding of the projective $n$-space $\mathbb {P}^n$ have the expected dimension, modulo a few well-known exceptions. It is arguably the first complete result on the dimensions of secant varieties of a classically studied variety since the work of Alexander and Hirschowitz in 1995. As Bernardi, Catalisano, Gimigliano, and Idá demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., $d=3$, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension $n$ of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of $\mathbb {P}^{79}$ in $\mathbb {P}^{88559}$.References
- Hirotachi Abo, On non-defectivity of certain Segre-Veronese varieties, J. Symbolic Comput. 45 (2010), no. 12, 1254–1269. MR 2733377, DOI 10.1016/j.jsc.2010.06.008
- Hirotachi Abo, Varieties of completely decomposable forms and their secants, J. Algebra 403 (2014), 135–153. MR 3166068, DOI 10.1016/j.jalgebra.2013.12.027
- Hirotachi Abo and Maria Chiara Brambilla, Secant varieties of Segre-Veronese varieties $\Bbb P^m\times \Bbb P^n$ embedded by $\scr O(1,2)$, Experiment. Math. 18 (2009), no. 3, 369–384. MR 2555705, DOI 10.1080/10586458.2009.10129051
- Hirotachi Abo and Maria Chiara Brambilla, On the dimensions of secant varieties of Segre-Veronese varieties, Ann. Mat. Pura Appl. (4) 192 (2013), no. 1, 61–92. MR 3011324, DOI 10.1007/s10231-011-0212-3
- 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 10.1090/S0002-9947-08-04725-9
- J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201–222. MR 1311347
- Enrique Arrondo and Alessandra Bernardi, On the variety parameterizing completely decomposable polynomials, J. Pure Appl. Algebra 215 (2011), no. 3, 201–220. MR 2729216, DOI 10.1016/j.jpaa.2010.04.008
- E. Ballico, On the secant varieties to the tangent developable of a Veronese variety, J. Algebra 288 (2005), no. 2, 279–286. MR 2146130, DOI 10.1016/j.jalgebra.2005.03.031
- A. Bernardi, M. V. Catalisano, A. Gimigliano, and M. Idà, Secant varieties to osculating varieties of Veronese embeddings of $\Bbb P^n$, J. Algebra 321 (2009), no. 3, 982–1004. MR 2488563, DOI 10.1016/j.jalgebra.2008.10.020
- Maria Chiara Brambilla and Giorgio Ottaviani, On the Alexander-Hirschowitz theorem, J. Pure Appl. Algebra 212 (2008), no. 5, 1229–1251. MR 2387598, DOI 10.1016/j.jpaa.2007.09.014
- Peter Bürgisser, Cook’s versus Valiant’s hypothesis, Theoret. Comput. Sci. 235 (2000), no. 1, 71–88. Selected papers in honor of Manuel Blum (Hong Kong, 1998). MR 1765966, DOI 10.1016/S0304-3975(99)00183-8
- Peter Bürgisser, Michael Clausen, and M. Amin Shokrollahi, Algebraic complexity theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 315, Springer-Verlag, Berlin, 1997. With the collaboration of Thomas Lickteig. MR 1440179, DOI 10.1007/978-3-662-03338-8
- M. V. Catalisano, L. Chiantini, A. V. Geramita, and A. Oneto, Waring-like decompositions of polynomials - $1$, preprint, arXiv:1511.07789.
- M. V. Catalisano, A. V. Geramita, and A. Gimigliano, On the secant varieties to the tangential varieties of a Veronesean, Proc. Amer. Math. Soc. 130 (2002), no. 4, 975–985. MR 1873770, DOI 10.1090/S0002-9939-01-06251-7
- M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Higher secant varieties of Segre-Veronese varieties, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 81–107. MR 2202248
- Luca Chiantini, Giorgio Ottaviani, and Nick Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1265–1287. MR 3270978, DOI 10.1137/140961389
- Luca Chiantini, Giorgio Ottaviani, and Nick Vannieuwenhoven, On generic identifiability of symmetric tensors of subgeneric rank, Trans. Amer. Math. Soc. 369 (2017), no. 6, 4021–4042. MR 3624400, DOI 10.1090/tran/6762
- Timothy A. Davis, Direct methods for sparse linear systems, Fundamentals of Algorithms, vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2006. MR 2270673, DOI 10.1137/1.9780898718881
- I. S. Duff, A. M. Erisman, and J. K. Reid, Direct methods for sparse matrices, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 892734
- Jean-Guillaume Dumas, Pascal Giorgi, and Clément Pernet, Dense linear algebra over word-size prime fields: the FFLAS and FFPACK packages, ACM Trans. Math. Software 35 (2008), no. 3, Art. 19, 35. MR 2738206, DOI 10.1145/1391989.1391992
- I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
- G. Guennebaud, B. Jacob, et al., Eigen v3, http://eigen.tuxfamily.org, 2010.
- Antonio Laface and Elisa Postinghel, Secant varieties of Segre-Veronese embeddings of $(\Bbb {P}^1)^r$, Math. Ann. 356 (2013), no. 4, 1455–1470. MR 3072808, DOI 10.1007/s00208-012-0890-1
- J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics, vol. 128, American Mathematical Society, Providence, RI, 2012. MR 2865915, DOI 10.1090/gsm/128
- J. M. Landsberg, Geometric complexity theory: an introduction for geometers, Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), no. 1, 65–117. MR 3343444, DOI 10.1007/s11565-014-0202-7
- W. Qian, Z. Xianyi, Z. Yunquan, and Q. Yi, AUGEM: Automatically generate high performance dense linear algebra kernels on x86 CPUs, Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis (New York, NY), ACM, 2013, pp. 25:1–25:12.
- Yong Su Shin, Secants to the variety of completely reducible forms and the Hilbert function of the union of star-configurations, J. Algebra Appl. 11 (2012), no. 6, 1250109, 27. MR 2997451, DOI 10.1142/S0219498812501095
- Amir Shpilka and Amir Yehudayoff, Arithmetic circuits: a survey of recent results and open questions, Found. Trends Theor. Comput. Sci. 5 (2009), no. 3-4, 207–388 (2010). MR 2756166, DOI 10.1561/0400000039
- A. Terracini, Sulla $V_k$ per cui la varietà degli $S_h$ $h+1$-secanti ha dimensione minore dell’ordinario, Rend. Circ. Mat. Palermo 31 (1911), 392–396.
- Douglas A. Torrance, Generic forms of low Chow rank, J. Algebra Appl. 16 (2017), no. 3, 1750047, 10. MR 3626714, DOI 10.1142/S0219498817500475
Additional Information
- Hirotachi Abo
- Affiliation: Department of Mathematics, University of Idaho, Moscow, Idaho 83844-1103
- MR Author ID: 614361
- Email: abo@uidaho.edu
- Nick Vannieuwenhoven
- Affiliation: Department of Computer Science, KU Leuven, B-3000 Leuven, Belgium
- MR Author ID: 977418
- ORCID: 0000-0001-5692-4163
- Email: nick.vannieuwenhoven@cs.kuleuven.be
- Received by editor(s): October 30, 2015
- Received by editor(s) in revised form: March 30, 2016
- Published electronically: August 3, 2017
- Additional Notes: The first author was partly supported by NSF grant DMS-0901816
The second author’s research was partially supported by a Ph.D. Fellowship of the Research Foundation–Flanders (FWO) and partially by a Postdoctoral Fellowship of the Research Foundation–Flanders (FWO) - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 393-420
- MSC (2010): Primary 14M99, 14Q15, 14Q20, 15A69, 15A72
- DOI: https://doi.org/10.1090/tran/6955
- MathSciNet review: 3717984