Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers

TL;DR

This paper develops new algebraic methods using liaison theory to analyze Jacobian ideals of hyperplane arrangements, establishing conditions under which their algebraic invariants depend solely on the intersection lattice, and providing new criteria for freeness.

Contribution

It introduces a novel approach using liaison theory to study Jacobian ideals, linking algebraic invariants to combinatorial data, and offers new freeness criteria for hyperplane arrangements.

Findings

Hilbert functions of Jacobian ideals are determined by intersection lattice under mild conditions

New bounds on the global Tjurina number of arrangements

Freeness characterized by minimal generators of related ideals

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.