Determinantal representation and subschemes of general plane curves

Let M=(mij) be an n×n square matrix of integers. For our purposes, we can assume without loss of generality that M is homogeneous and that the entries are non-increasing going leftward and downward. Let d be the sum of the entries on either diagonal. We give a complete characterization of which such matrices have the property that a general form of degree d in C[x0,x1,x2] can be written as the determinant of a matrix of forms (fij) with degfij=mij (of course fij=0ifmij<0). As a consequence, we answer the related question of which (n-1)×n matrices Q of integers have the property that a general plane curve of degree d contains a zero-dimensional subscheme whose degree Hilbert-Burch matrix is Q. This leads to an algorithmic method to determine properties of linear series contained in general plane curves.