Jacobian Schemes Arising From Hypersurface Arrangements in ℙn

Abstract

Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in ${\mathbb{P}}^{n}$ is free if $S/J$ is Cohen–Macaulay (CM), where $S = K[x_{0},\ldots ,x_{n}]$ and $J$ is the Jacobian ideal. We study three related unmixed ideals: $J^{top}$, the intersection of height two primary components, $\sqrt{J^{top}}$, the radical of $J^{top}$, and when the $f_{i}$ are smooth we also study $\sqrt{J}$. Under mild hypotheses, we show that these ideals are CM. This establishes a full generalization of an earlier result with Schenck from hyperplane arrangements to hypersurface arrangements. If the hypotheses fail for an arrangement in projective $3$-space, the Hartshorne–Rao module measures the failure of CMness. We establish consequences for the even liaison classes of $J^{top}$ and $\sqrt{J}$.