2025/08/16 by Juan Migliore, Migliore, Juan, Uwe Nagel +1
Computer Science · Engineering · Mathematics · #13C40 #14N20 #32S22 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2508.12113
openalex publication_date 2025/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A hyperplane arrangement \cA is said to be free if the corresponding Jacobian ideal J_\cA is Cohen-Macaulay. If \cA is free then J_\cA is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not well understood. A conjecture of Terao says that freeness of \cA depends only on the intersection lattice of \cA. Given an arrangement \cA, we define the ideal J_\cAtop to be the intersection of the codimension 2 primary components of J_\cA. This ideal is unmixed, but not necessarily Cohen-Macaulay; if \cA is free then J_\cA = J_\cAtop. We develop a new method for studying the ideals J_\cA and J_\cAtop and establish results in the spirit of Terao's conjecture, focusing on J_\cAtop rather than J_\cA. It is based on a new application of liaison theory, the general residual of \cA. This residual ideal defines a scheme with surprisingly simple properties. These allow us to track back to J_\cAtop. Extending earlier results with Schenck, we identify mild conditions on a hyperplane arrangement which imply that the Hilbert function of \Jac( f_\cA)top or even its graded Betti numbers, are determined by the intersection lattice of \cA. We establish new bounds on the global Tjurina number of a hyperplane arrangement. For line arrangements, we show that the graded Betti numbers of \Jac( f_\cA)sat determine the graded Betti numbers of \Jac( f_\cA), and of the corresponding Milnor module J_\cAsat/J_\cA. We obtain a new freeness criterion for line arrangements -- it highlights the fact that free line arrangements are special by proving that a related codimension two ideal has the least possible number of generators, namely two, if and only if \cA is free. We illustrate our results by computing the graded Betti numbers for a number of basic arrangements that were not accessible with previous methods.