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arXiv · 2610.28792

Closedness of bounded strength and partition rank loci

Benjamin Baily·Alessandro Danelon
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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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Benjamin Baily, Alessandro Danelon. 2026-10-20. Closedness of bounded strength and partition rank loci. https://arxiv.org/abs/2610.28792

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