On the matrices of given rank in a large subspace

Let V be a linear subspace of Mn,p(K) with codimension lesser than n, where K is an arbitrary field and n⩾p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K≃F2. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r∈〚1,p-1〛. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.