Spaces of matrices of constant rank and uniform vector bundles

We consider the problem of determining l(r,a)l(r,a), the maximal dimension of a subspace of a×aa×a matrices of rank r  . We first review, in the language of vector bundles, the known results. Then using known facts on uniform bundles we prove some new results and make a conjecture. Finally we determine l(r;a)l(r;a) for every r  , 1≤r≤a1≤r≤a, when a≤10a≤10, showing that our conjecture holds true in this range.