Jacobian schemes arising from hypersurface arrangements in \mathbb Pⁿ

2023/12/02 by Juan Migliore, Migliore, Juan, Uwe Nagel +1
Mathematics · #13C40 #13H10 #14M05 #14M06 #14N20 #32S22 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2312.01192

openalex publication_date 2023/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Freeness is an important property of a hypersurface arrangement, although its presence is not well understood. A hypersurface arrangement in \PPⁿ 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 f_i 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 J^top and √(J).

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