ALGEBRAIC COMBINATORICS

no. 5
A characteristic free approach to secant varieties of triple Segre products
Conca, Aldo1; De Negri, Emanuela1; Stojanac, Željka2
1 Dipartimento di Matematica Università di Genova via Dodecaneso 35 Genova Italy
2 Institute for Theoretical Physics University of Cologne Germany
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 ℙ 1 ×ℙ a-1 ×ℙ 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 [] 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:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.115
Classification: 13C40, 05E40, 13P20
Keywords: Segre products, secant varieties, Gröbner bases, tensors.
Conca, Aldo 1; De Negri, Emanuela 1; Stojanac, Željka 2

1 Dipartimento di Matematica Università di Genova via Dodecaneso 35 Genova Italy
2 Institute for Theoretical Physics University of Cologne Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2020__3_5_1011_0,
     author = {Conca, Aldo and De Negri, Emanuela and Stojanac, \v{Z}eljka},
     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 = {https://alco.centre-mersenne.org/articles/10.5802/alco.115/}
}
TY  - JOUR
AU  - Conca, Aldo
AU  - De Negri, Emanuela
AU  - Stojanac, Željka
TI  - A characteristic free approach to secant varieties of triple Segre products
JO  - Algebraic Combinatorics
PY  - 2020
SP  - 1011
EP  - 1021
VL  - 3
IS  - 5
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.115/
DO  - 10.5802/alco.115
LA  - en
ID  - ALCO_2020__3_5_1011_0
ER  - 
%0 Journal Article
%A Conca, Aldo
%A De Negri, Emanuela
%A Stojanac, Željka
%T A characteristic free approach to secant varieties of triple Segre products
%J Algebraic Combinatorics
%D 2020
%P 1011-1021
%V 3
%N 5
%I MathOA foundation
%U https://alco.centre-mersenne.org/articles/10.5802/alco.115/
%R 10.5802/alco.115
%G en
%F 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/articles/10.5802/alco.115/

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

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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) | DOI | MR | Zbl

Cited by Sources: