Dimension of tensor loci with prescribed minimum border subrank
Let $\mathbb{F}$ be an algebraically closed field, $k \ge 3$, and $V = \mathbb{F}^{n_1} \otimes \cdots \otimes \mathbb{F}^{n_k}$. The border subrank $\underline{Q}(T)$ of $T \in V$ is the largest $r$ such that $\sum_{a=1}^r e_a^{\otimes k}$ lies in the Zariski closure of $\{(L_1 \otimes \cdots \otimes L_k)T : L_j \in \operatorname{Hom}(\mathbb{F}^{n_j}, \mathbb{F}^r)\}$. For $1 \le r \le \min_j n_j$ let $$X_r^{(n_1, \dots, n_k)} = \{T \in V : \underline{Q}(T) \ge r\},$$ whose dimension is that of its Zariski closure. Biaggi, Chang, Draisma and Rupniewski proved that this set is constructible and bounded its dimension from above; they also showed that a general tensor in $(\mathbb{F}^n)^{\otimes 3}$ has border subrank at least $\lfloor \sqrt{4n} \rfloor - 3$. Since border subrank is at least subrank, Pielasa, Šafránek and Shatsila's formula for the generic subrank gives $\dim X_r^{(n_1, \dots, n_k)} = n_1 \cdots n_k$ for $r \le \min\{n_1, \dots, n_k, \lfloor (n_1 + \cdots + n_k - k + 1)^{1/(k-1)} \rfloor\}$. Chang gave a lower bound on the dimension over $\mathbb{C}$ for $k = 3$ and $r = n_1 = n_2 = n_3$.
Problem. Determine $\dim X_r^{(n_1, \dots, n_k)}$ for every algebraically closed field $\mathbb{F}$, every $k \ge 3$, all positive integers $n_1, \dots, n_k$ and every $1 \le r \le \min_j n_j$.
Source: Oneto and Ventura (2025), Question 14, over an infinite field; stated over algebraically closed fields by OpenProblemsInNLA.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- border subranks, OpenProblemsInNLA, tensor degenerations
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{oneto2025,
author = {Oneto, Alessandro and Ventura, Emanuele},
title = {Ranks of tensors: geometry and applications},
journal = {Boll. Unione Mat. Ital.},
volume = {18},
number = {3},
year = {2025},
pages = {751--778},
doi = {10.1007/s40574-025-00472-9},
}
@article{biaggi2025,
author = {Biaggi, Benjamin and Chang, Chia-Yu and Draisma, Jan and Rupniewski, Filip},
title = {Border subrank via a generalised {H}ilbert--{M}umford criterion},
journal = {Adv. Math.},
volume = {461},
year = {2025},
pages = {110077},
doi = {10.1016/j.aim.2024.110077},
eprint = {2402.10674},
archivePrefix = {arXiv},
}
@article{pielasa2025,
author = {Pielasa, Pawe{\l} and {\v{S}}afr{\'a}nek, Matou{\v{s}} and Shatsila, Anatoli},
title = {Exact values of generic subrank},
journal = {Adv. Math.},
volume = {469},
year = {2025},
pages = {110234},
doi = {10.1016/j.aim.2025.110234},
eprint = {2408.07550},
archivePrefix = {arXiv},
}
@article{chang2025,
author = {Chang, Chia-Yu},
title = {Maximal border subrank tensors},
journal = {Linear Multilinear Algebra},
volume = {73},
number = {3},
year = {2025},
pages = {525--535},
doi = {10.1080/03081087.2024.2352456},
eprint = {2208.04281},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr30,
author = {Townsend, Alex},
title = {{TR-30}: Dimension of tensor loci with prescribed minimum border subrank},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-30/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-00138, "Dimension of tensor loci with prescribed minimum border subrank", clopen.solutions.