Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers

Abstract

A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ 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 $\cA$ depends only on the intersection lattice of $\cA$. Given an arrangement $\cA$, we define the ideal $J_\cA^{top}$ to be the intersection of the codimension 2 primary components of $J_\cA$. This ideal is unmixed, but not necessarily Cohen-Macaulay; if $\cA$ is free then $J_\cA = J_\cA^{top}$. We develop a new method for studying the ideals $J_\cA$ and $J_\cA^{top}$ and establish results in the spirit of Terao's conjecture, focusing on $J_\cA^{top}$ rather than $J_\cA$. It is based on a new application of liaison theory, the general residual of $\cA$. This residual ideal defines a scheme with surprisingly simple properties. These allow us to track back to $J_\cA^{top}$. Extending earlier results with Schenck, we identify mild conditions on a hyperplane arrangement which imply that the Hilbert function of $\Jac( f_\cA)^{top}$ or even its graded Betti numbers, are determined by the intersection lattice of $\cA$. We establish new bounds on the global Tjurina number of a hyperplane arrangement. For line arrangements, we show that the graded Betti numbers of $\Jac( f_\cA)^{sat}$ determine the graded Betti numbers of $\Jac( f_\cA)$, and of the corresponding Milnor module $J_\cA^{sat}/J_\cA$. 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 $\cA$ 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.