Linear sections of determinantal varieties

Denote Mmn(k) to be the set of all m n matrices over k, and PMmn(k) to be its affine space. Let Dm,n,r PMmn(k) be the subvariety consisting of the projective classes of all m n matrices with rank less than or equal to r. We study the minimal integer k > 0 such that for any linear subspace L of dimension k in PMmn(k), Dm,n,r L = . We get some partial results about this problem. Specially, when m = n = r + 1, we solve this problem for k = R, C and Q.