Jacobian Schemes Arising From Hypersurface Arrangements in ℙn

Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in Pⁿ is free if $S/J$ is Cohen–Macaulay (CM), where S = K[x₀,… ,xₙ] and J is the Jacobian ideal. We study three related unmixed ideals: Jᵗᵒᵖ, the intersection of height two primary components, √Jᵗᵒᵖ, the radical of Jᵗᵒᵖ, and when the fᵢ are smooth we also study √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 Jᵗᵒᵖ and √J.