Schemes Supported on the Singular Locus of a Hyperplane Arrangement in ℙn

Abstract

Abstract A hyperplane arrangement in $\mathbb P^n$ is free if $R/J$ is Cohen–Macaulay (CM), where $R = k[x_0,\dots ,x_n]$ and $J$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $ J^{un}$, the intersection of height two primary components, and $\sqrt{J}$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $\mathbb P^3$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $r$, there is an arrangement for which $R/J^{un}$ (resp. $R/\sqrt{J}$) fails to be CM in only one degree, and this failure is by $r$. We draw consequences for the even liaison class of $J^{un}$ or $\sqrt{J}$.