Spaces of matrices of constant rank and uniform vector bundles

TL;DR

This paper investigates the maximal dimension of subspaces of endomorphisms with constant rank, connecting the problem to vector bundle theory, presenting new results, and proposing a conjecture.

Contribution

It introduces new bounds and insights into the dimension of constant rank subspaces of endomorphisms using vector bundle techniques.

Findings

Reviewed known results in the context of vector bundles

Proved new bounds for the dimension of rank r subspaces

Formulated a conjecture on the maximal dimension

Abstract

Let A be a k-vector space of dimension a. A subvector space M of End(A) is said to be of rank r if every non-zero f in M has rank r. The problem considered in this paper is to determine l(r;a) the maximal dimension of a rank r subspace of End(A). Known results are reviewed in the language of vector bundles. Some new results are proved and a conjecture is made.