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Closedness of bounded strength and partition rank loci
Abstract
We show the existence of a quartic form of strength $3$ in the closure of the strength at most $2$ locus, answering a question of Ballico--Bik--Oneto--Ventura in the negative. In the analogous setting of partition rank as introduced by Naslund, we further show that the locus of tensors of partition rank at most $2$ is closed, but the locus of tensors of partition rank at most $3$ is not. Together with a suspension lemma, this answers for all r whether the locus of forms/tensors of strength/partition rank at most r is Zariski-closed.
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BibTeX
@misc{baily2026closedness,
title = {{Closedness of bounded strength and partition rank loci}},
author = {Benjamin Baily and Alessandro Danelon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28792},
eprint = {2610.28792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28792},
}
RIS
TY - UNPB
TI - Closedness of bounded strength and partition rank loci
AU - Benjamin Baily
AU - Alessandro Danelon
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28792
UR - https://arxiv.org/abs/2610.28792
AB - We show the existence of a quartic form of strength $3$ in the closure of the strength at most $2$ locus, answering a question of Ballico--Bik--Oneto--Ventura in the negative. In the analogous setting of partition rank as introduced by Naslund, we further show that the locus of tensors of partition rank at most $2$ is closed, but the locus of tensors of partition rank at most $3$ is not. Together with a suspension lemma, this answers for all r whether the locus of forms/tensors of strength/partition rank at most r is Zariski-closed.
ER -