Geometric Rank and Linear Determinantal Varieties

TL;DR

This paper explores the relationship between geometric ranks of tensors and linear determinantal varieties, providing bounds, classifications, and extending results to tensors with multiple parts.

Contribution

It introduces bounds on dimensions of determinantal varieties, classifies tensors with specific geometric ranks, and extends tripartite tensor results to n-part tensors.

Findings

Classified tensors with geometric rank 3.

Established upper bounds for multilinear ranks of tensors with geometric rank 4.

Proved the equivalence between geometric rank 1 and partition rank 1 in n-part tensors.

Abstract

There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition rank 1.