A characteristic free approach to secant varieties of triple Segre products
Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1011-1021.

The goal of this short note is to study the secant varieties of the Segre embedding of the product 1 × a-1 × b-1 by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic the results of Landsberg and Weyman [5] regarding the defining ideal and the Cohen–Macaulay property of the secant varieties. Furthermore we compute their degrees and give a bound for their Castelnuovo–Mumford regularities, which are sharp in many cases.

Received: 2019-07-29
Revised: 2020-02-10
Accepted: 2020-02-12
Published online: 2020-10-12
DOI: https://doi.org/10.5802/alco.115
Classification: 13C40,  05E40,  13P20
Keywords: Segre products, secant varieties, Gröbner bases, tensors.
@article{ALCO_2020__3_5_1011_0,
     author = {Conca, Aldo and De Negri, Emanuela and Stojanac, \v Zeljka},
     title = {A characteristic free approach to secant varieties of triple Segre products},
     journal = {Algebraic Combinatorics},
     pages = {1011--1021},
     publisher = {MathOA foundation},
     volume = {3},
     number = {5},
     year = {2020},
     doi = {10.5802/alco.115},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_5_1011_0/}
}
Conca, Aldo; De Negri, Emanuela; Stojanac, Željka. A characteristic free approach to secant varieties of triple Segre products. Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1011-1021. doi : 10.5802/alco.115. https://alco.centre-mersenne.org/item/ALCO_2020__3_5_1011_0/

[1] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, Volume 39, Cambridge University Press, Cambridge, 1993, xii+403 pages | MR 1251956 | Zbl 0788.13005

[2] Gessel, Ira; Viennot, Gérard Binomial determinants, paths, and hook length formulae, Adv. Math., Volume 58 (1985) no. 3, pp. 300-321 | Article | MR 815360 | Zbl 0579.05004

[3] Kalkbrener, Michael; Sturmfels, Bernd Initial complexes of prime ideals, Adv. Math., Volume 116 (1995) no. 2, pp. 365-376 | Article | MR 1363769 | Zbl 0877.13025

[4] Landsberg, Joseph M. Tensors: geometry and applications, Graduate Studies in Mathematics, Volume 128, American Mathematical Society, Providence, RI, 2012, xx+439 pages | MR 2865915 | Zbl 1238.15013

[5] Landsberg, Joseph M.; Weyman, Jerzy On the ideals and singularities of secant varieties of Segre varieties, Bull. Lond. Math. Soc., Volume 39 (2007) no. 4, pp. 685-697 | Article | MR 2346950 | Zbl 1130.14041

[6] Mirsky, Leon A dual of Dilworth’s decomposition theorem, Amer. Math. Monthly, Volume 78 (1971), p. 876-877 | Article | MR 288054 | Zbl 0263.06002

[7] Oeding, Luke Are all secant varieties of Segre products arithmetically Cohen–Macaulay? (2016) (https://arxiv.org/abs/1603.08980)

[8] Simis, Aron; Ulrich, Bernd On the ideal of an embedded join, J. Algebra, Volume 226 (2000) no. 1, pp. 1-14 | Article | MR 1749874 | Zbl 1034.14026

[9] Sturmfels, Bernd; Sullivant, Seth Combinatorial secant varieties, Pure Appl. Math. Q., Volume 2 (2006) no. 3, pp. 867-891 (Special Issue: In honor of Robert D. MacPherson. Part 1) | Article | MR 2252121 | Zbl 1107.14045