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, . . . , x_n] and J is the Jacobian ideal. We study three related unmixed ideals: J^top, the intersection of height two primary components, J√^top, the radical of J^top, and when the fi are smooth we also study vJ. 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 v√J.