Cohen–Macaulay coordinate rings for every tensor border-rank variety
For $k \ge 3$ and $n_1, \dots, n_k \ge 1$ let $V = \mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_k}$ and, for $r \ge 1$, let $X_r \subset V$ be the variety of tensors of border rank at most $r$, the affine cone over the $r$-th secant variety of the Segre variety $\mathbb{P}^{n_1 - 1} \times \cdots \times \mathbb{P}^{n_k - 1}$. Let $A_r = \mathbb{C}[V]/I(X_r)$ be its coordinate ring. Landsberg and Weyman proved that $A_2$ is Cohen–Macaulay when $k = 4$, and $A_3$ when $k = 3$ and $n_1, n_2, n_3 \ge 3$. Oeding proved that $A_4$ is Cohen–Macaulay when $k = 3$, $n_1 = n_2 = 3$ and $n_3 \ge 4$.
Problem. Is $A_r$ Cohen–Macaulay for every $k \ge 3$, all $n_1, \dots, n_k \ge 1$ and every $r \ge 1$?
Source: Oeding (2016), Conjecture 1.1, after a question of Landsberg and Weyman (2007), §1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- AC, NA
- Keywords
- Cohen–Macaulay rings, OpenProblemsInNLA, secant varieties, Segre varieties
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@misc{oeding2016,
author = {Oeding, Luke},
title = {Are all secant varieties of {S}egre products arithmetically {C}ohen--{M}acaulay?},
year = {2016},
eprint = {1603.08980},
archivePrefix = {arXiv},
}
@article{landsberg2007,
author = {Landsberg, J. M. and Weyman, Jerzy},
title = {On the ideals and singularities of secant varieties of {S}egre varieties},
journal = {Bull. Lond. Math. Soc.},
volume = {39},
number = {4},
year = {2007},
pages = {685--697},
doi = {10.1112/blms/bdm049},
eprint = {math/0601452},
archivePrefix = {arXiv},
}
@misc{jagiella2026,
author = {Jagie{\l}{\l}a, Jakub and Jelisiejew, Joachim},
title = {Unrestrictions and concise secant varieties},
year = {2026},
eprint = {2604.24879},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr25,
author = {Townsend, Alex},
title = {{TR-25}: Cohen–Macaulay coordinate rings for every tensor border-rank variety},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-25/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00135, "Cohen–Macaulay coordinate rings for every tensor border-rank variety", clopen.solutions.