Large affine spaces of matrices with rank bounded below

TL;DR

This paper classifies affine subspaces of matrices over a field with minimal codimension that contain only matrices of rank at least r, extending previous bounds and classifications.

Contribution

It provides a classification of affine matrix subspaces with minimal codimension and rank constraints, building on recent classifications of subspaces within GL_r(K).

Findings

Minimal codimension of such subspaces is (r+1)r/2.

Classification up to equivalence of these minimal codimension subspaces.

Utilizes recent classification of affine subspaces in GL_r(K).

Abstract

Let K be an arbitrary (commutative) field with at least three elements, and let n, p and r be positive integers with r<=min(n,p). In a recent work, we have proved that an affine subspace of M_{n,p}(K) containing only matrices of rank greater than or equal to r must have a codimension greater than or equal to (r+1)r/2. Here, we classify, up to equivalence, these subspaces with the minimal codimension (r+1)r/2. This uses our recent classification of the affine subspaces of M_r(K) contained in GL_r(K) and which have the maximal dimension r(r-1)/2.