A hyperplane arrangement is said to be free if the corresponding Jacobian ideal J_ is Cohen-Macaulay. If is free then J_ is unmixed (i. e. equidimensional). Freeness is an important property, yet its presence is not well understood. A conjecture of Terao says that freeness of depends only on the intersection lattice of. Given an arrangement, we define the ideal J_^top to be the intersection of the codimension 2 primary components of J_. This ideal is unmixed, but not necessarily Cohen-Macaulay; if is free then J_ = J_^top. We develop a new method for studying the ideals J_ and J_^top and establish results in the spirit of Terao's conjecture, focusing on J_^top rather than J_. It is based on a new application of liaison theory, the general residual of. This residual ideal defines a scheme with surprisingly simple properties. These allow us to track back to J_^top. Extending earlier results with Schenck, we identify mild conditions on a hyperplane arrangement which imply that the Hilbert function of (f_) ^top or even its graded Betti numbers, are determined by the intersection lattice of. We establish new bounds on the global Tjurina number of a hyperplane arrangement. For line arrangements, we show that the graded Betti numbers of (f_) ^sat determine the graded Betti numbers of (f_), and of the corresponding Milnor module J_^sat/J_. We obtain a new freeness criterion for line arrangements -- it highlights the fact that free line arrangements are special by proving that a related codimension two ideal has the least possible number of generators, namely two, if and only if is free. We illustrate our results by computing the graded Betti numbers for a number of basic arrangements that were not accessible with previous methods.
Migliore et al. (Sat,) studied this question.