The four-exception conjecture for alternating tensor border-rank varieties
For $p \ge 3$ and $N \ge 2p$ let $C_{p,N} \subset \bigwedge^p \mathbb{C}^N$ be the cone of decomposable vectors $v_1 \wedge \cdots \wedge v_p$, the affine cone over the Grassmannian $\mathrm{Gr}(p, N)$ in its Plücker embedding. For $s \ge 1$ let $Y_s$ be the Zariski closure of $\{x_1 + \cdots + x_s : x_i \in C_{p,N}\}$, whose expected dimension is $$E_{p,N,s} = \min\left\{\binom{N}{p},\ s\bigl(p(N-p)+1\bigr)\right\}.$$ For $p = 2$ the $Y_s$ are varieties of skew-symmetric matrices of bounded rank, and $\mathrm{Gr}(p, N) \cong \mathrm{Gr}(N-p, N)$ covers $N < 2p$. Catalisano, Geramita and Gimigliano proved that $\dim Y_s = E_{p,N,s}$ for $s \le N/p$, and that $(p, N, s) = (3, 7, 3)$, $(4, 8, 3)$ and $(3, 9, 4)$ are exceptions. Abo, Ottaviani and Peterson proved that $\dim Y_s = E_{p,N,s}$ for $s \le 6$ outside the four cases below, and Boralevi extended this to $s \le 12$. Baur, Draisma and de Graaf checked the statement below by computer for $N \le 15$.
Problem. Is $\dim Y_s = E_{p,N,s}$ for all $p \ge 3$, $N \ge 2p$ and $s \ge 1$, except exactly for the following four triples, where $\dim Y_s$ is as listed? $$\begin{array}{c|c} (p, N, s) & \dim Y_s \\ \hline (3, 7, 3) & 34 \\ (4, 8, 3) & 50 \\ (4, 8, 4) & 64 \\ (3, 9, 4) & 74 \end{array}$$
Source: Baur, Draisma and de Graaf (2007), Conjecture 4.1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- AG — Algebraic Geometry
- Secondary
- NA
- Keywords
- alternating tensors, Grassmannians, OpenProblemsInNLA, secant varieties
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{baur2007,
author = {Baur, Karin and Draisma, Jan and de Graaf, Willem A.},
title = {Secant dimensions of minimal orbits: computations and conjectures},
journal = {Experiment. Math.},
volume = {16},
number = {2},
year = {2007},
pages = {239--250},
doi = {10.1080/10586458.2007.10128997},
}
@article{catalisano2005,
author = {Catalisano, M. V. and Geramita, A. V. and Gimigliano, A.},
title = {Secant varieties of {G}rassmann varieties},
journal = {Proc. Amer. Math. Soc.},
volume = {133},
number = {3},
year = {2005},
pages = {633--642},
doi = {10.1090/S0002-9939-04-07632-4},
eprint = {math/0208166},
archivePrefix = {arXiv},
}
@article{abo2012,
author = {Abo, Hirotachi and Ottaviani, Giorgio and Peterson, Chris},
title = {Non-defectivity of {G}rassmannians of planes},
journal = {J. Algebraic Geom.},
volume = {21},
number = {1},
year = {2012},
pages = {1--20},
doi = {10.1090/S1056-3911-2010-00540-1},
eprint = {0901.2601},
archivePrefix = {arXiv},
}
@article{boralevi2013,
author = {Boralevi, Ada},
title = {A note on secants of {G}rassmannians},
journal = {Rend. Istit. Mat. Univ. Trieste},
volume = {45},
year = {2013},
pages = {67--72},
url = {https://rendiconti.dmi.units.it/volumi/45/011.pdf},
}
@misc{blomenhofer2023,
author = {Taveira Blomenhofer, Alexander and Casarotti, Alex},
title = {Nondefectivity of invariant secant varieties},
year = {2023},
eprint = {2312.12335},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-tr31,
author = {Townsend, Alex},
title = {{TR-31}: The four-exception conjecture for alternating tensor border-rank varieties},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-31/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-00139, "The four-exception conjecture for alternating tensor border-rank varieties", clopen.solutions.