On degenerate geometric rank

Matěj Doležálek; Paweł Pielasa; Derek Wu

On degenerate geometric rank

Algebraic Geometry 2026-07-21 v1

Authors: Matěj Doležálek , Paweł Pielasa , Derek Wu

Abstract

A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.

Keywords

degenerate geometric rank tensor linear space of matrices rank locus lifting

Cite

@article{arxiv.2607.18898,
  title  = {On degenerate geometric rank},
  author = {Matěj Doležálek and Paweł Pielasa and Derek Wu},
  journal= {arXiv preprint arXiv:2607.18898},
  year   = {2026}
}

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10 pages