Algebra & Number Theory

Secant varieties of Segre–Veronese varieties

Claudiu Raicu

Full-text: Open access

Abstract

We prove that the ideal of the variety of secant lines to a Segre–Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre–Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.

Article information

Source
Algebra Number Theory, Volume 6, Number 8 (2012), 1817-1868.

Dates
Received: 30 June 2011
Revised: 15 December 2011
Accepted: 20 January 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729916

Digital Object Identifier
doi:10.2140/ant.2012.6.1817

Mathematical Reviews number (MathSciNet)
MR3033528

Zentralblatt MATH identifier
1273.14102

Subjects
Primary: 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 14M12: Determinantal varieties [See also 13C40]

Keywords
Segre varieties Veronese varieties secant varieties

Citation

Raicu, Claudiu. Secant varieties of Segre–Veronese varieties. Algebra Number Theory 6 (2012), no. 8, 1817--1868. doi:10.2140/ant.2012.6.1817. https://projecteuclid.org/euclid.ant/1513729916


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References

  • E. S. Allman, “Open problem: Determine the ideal defining $Sec_4( \mathbb{P}^3\times\mathbb{P}^3\times \mathbb{P}^3)$”, paper, 2007, http://www.dms.uaf.edu/~eallman/Papers/salmonPrize.pdf.
  • E. S. Allman and J. A. Rhodes, “Phylogenetic ideals and varieties for the general Markov model”, Adv. in Appl. Math. 40:2 (2008), 127–148.
  • D. J. Bates and L. Oeding, “Toward a salmon conjecture”, Exp. Math. 20:3 (2011), 358–370.
  • A. Bernardi, “Ideals of varieties parameterized by certain symmetric tensors”, J. Pure Appl. Algebra 212:6 (2008), 1542–1559.
  • D. Cartwright, D. Erman, and L. Oeding, “Secant varieties of $\mathbb P^2\times\mathbb P^n$ embedded by $\mathscr O(1,2)$”, J. Lond. Math. Soc. $(2)$ 85:1 (2012), 121–141.
  • M. V. Catalisano, A. V. Geramita, and A. Gimigliano, “On the ideals of secant varieties to certain rational varieties”, J. Algebra 319:5 (2008), 1913–1931.
  • D. Cox and J. Sidman, “Secant varieties of toric varieties”, J. Pure Appl. Algebra 209:3 (2007), 651–669.
  • J. Draisma and J. Kuttler, “Bounded-rank tensors are defined in bounded degree”, preprint, 2011.
  • L. Ein and R. Lazarsfeld, “Asymptotic syzygies of algebraic varieties”, Invent. Math. 190:3 (2012), 603–646.
  • S. Friedland, “On tensors of border rank $l$ in $\mathbb{C}^{m\times n\times l}$”, preprint, 2010. \codarefarXiv 1003.1968
  • S. Friedland and E. Gross, “A proof of the set-theoretic version of the salmon conjecture”, J. Algebra 356 (2012), 374–379.
  • W. Fulton and J. Harris, Representation theory: A first course, Graduate Texts in Mathematics 129, Springer, New York, 1991.
  • L. D. Garcia, M. Stillman, and B. Sturmfels, “Algebraic geometry of Bayesian networks”, J. Symbolic Comput. 39:3-4 (2005), 331–355.
  • D. R. Grayson and M. E. Stillman, Macaulay 2: A software system for research in algebraic geometry, 1993, http://www.math.uiuc.edu/Macaulay2/.
  • R. Grone, “Decomposable tensors as a quadratic variety”, Proc. Amer. Math. Soc. 64:2 (1977), 227–230.
  • H. T. Hà, “Box-shaped matrices and the defining ideal of certain blowup surfaces”, J. Pure Appl. Algebra 167:2-3 (2002), 203–224.
  • V. Kanev, “Chordal varieties of Veronese varieties and catalecticant matrices”, J. Math. Sci. $($New York$)$ 94:1 (1999), 1114–1125.
  • J. M. Landsberg, Tensors: geometry and applications, Graduate Studies in Mathematics 128, American Mathematical Society, Providence, RI, 2012.
  • J. M. Landsberg and L. Manivel, “On the ideals of secant varieties of Segre varieties”, Found. Comput. Math. 4:4 (2004), 397–422.
  • J. M. Landsberg and L. Manivel, “Generalizations of Strassen's equations for secant varieties of Segre varieties”, Comm. Algebra 36:2 (2008), 405–422.
  • J. M. Landsberg and J. Weyman, “On the ideals and singularities of secant varieties of Segre varieties”, Bull. Lond. Math. Soc. 39:4 (2007), 685–697.
  • J. M. Landsberg and J. Weyman, “On secant varieties of compact Hermitian symmetric spaces”, J. Pure Appl. Algebra 213:11 (2009), 2075–2086.
  • L. Manivel, “On spinor varieties and their secants”, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 078, 22.
  • L. Oeding and C. Raicu, “Tangential varieties of Segre varieties”, preprint, 2011.
  • L. Pachter and B. Sturmfels, “Tropical geometry of statistical models”, Proc. Natl. Acad. Sci. USA 101:46 (2004), 16132–16137.
  • M. Pucci, “The Veronese variety and catalecticant matrices”, J. Algebra 202:1 (1998), 72–95.
  • C. Raicu, “$3\times 3$ minors of catalecticants”, preprint, 2010.
  • C. C. Raicu, Secant varieties of Segre–Veronese varieties, thesis, University of California, Berkeley, 2011, https://web.math.princeton.edu/~craicu/thesis.pdf.
  • A. Snowden, “Syzygies of Segre embeddings”, preprint, 2010.
  • E. K. Wakeford, “On canonical forms”, Proc. London Math. Soc. 18:2 (1919), 403–410.
  • J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics 149, Cambridge University Press, 2003.