Freeness classification and non-free loci of determinantal arrangements

Determine whether the determinantal divisor D_{m,n}, defined as the union of the zero loci of the maximal minors of a generic m-by-n complex matrix with 1 <= m < n, is free if and only if m <= 2 or n = m+1; additionally, prove that for 3 <= m <= n-2 its non-free locus contains every matrix of rank at most m-1, and that for m = n-2 the non-free locus is exactly the variety of matrices of rank at most n-3.

Background

The paper studies the divisor D_{m,n} formed by the product of all maximal minors of a generic m-by-n matrix. It establishes freeness in the families m=1, m=2, and n=m+1, while proving non-freeness for D_{3,5}, D_{3,6}, D_{4,6}, and D_{5,7}. These computations motivate a general classification conjecture.

The conjecture also specifies the expected singular behavior: in the intermediate range 3 <= m <= n-2, all matrices of rank at most m-1 should lie in the non-free locus, while in the dual family m=n-2 the non-free locus should be precisely the rank-at-most-(n-3) determinantal variety. The paper verifies the latter description for D_{3,5} and establishes freeness at full-rank points for the relevant cases, but does not prove the general assertion.