On defective spans of singular vector tuples beyond the boundary format

TL;DR

This paper investigates the properties of tensor spaces beyond the boundary format, revealing defective behaviors in singular vector tuples and relating these to cohomological dimensions and Koszul cohomology.

Contribution

It introduces a new analysis of critical spaces in tensor spaces beyond the boundary format and identifies an infinite family of tensors with defective singular vector span behavior.

Findings

Identifies defective behavior in singular vector tuples for certain tensor spaces.

Relates the codimension of the span to the kernel of a cohomology map.

Proposes a conjecture on the classification of critical space behavior.

Abstract

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an arbitrary number of factors, we relate the codimension of this span within the critical space to the dimension of the kernel of a map in cohomology. Furthermore, we exhibit an infinite family of order-three tensors with a defective behavior: the general span of singular vector tuples achieves the maximum possible codimension rather than the expected minimum. Finally, we conjecture a classification of the behavior of critical spaces in this regime and draw a connection to Koszul cohomology.