Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers

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.