2019/08/11 by Juan Migliore, Migliore, J., U. Nagel +3
Mathematics · #Commutative Algebra and Its Applications #Algebraic Geometry and Number Theory #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1908.03939
We introduce the use of liaison addition to the study of hyperplane arrangements. For an arrangement, \mathcal A, of hyperplanes in \mathbb Pn, \mathcal A is free if R/J is Cohen-Macaulay, where J is the Jacobian ideal of \mathcal A. Terao's conjecture says that freeness of \mathcal A is determined by the combinatorics of the intersection lattice of \mathcal A. We study the Cohen-Macaulayness of three other ideals, all unmixed, that are closely related to \mathcal A. Let J = \mathfrak q1 ∩ … ∩ \mathfrak qs be the intersection of height two primary components of J and √(J) = \mathfrak p1 ∩ … ∩ \mathfrak ps be the radical of J. Our third ideal is \mathfrak p1b1 ∩ … ∩ \mathfrak psbs for suitable b1,…, bs. With a mild hypothesis we use liaison addition to show that all of these ideals are Cohen-Macaulay. When our hypothesis does not hold, we show that these ideals are not necessarily Cohen-Macaulay, and that Cohen-Macaulayness of any of these ideals does not imply Cohen-Macaulayness of any of the others. While we do not study the freeness of \mathcal A, we show by example that the Betti diagrams can vary even for arrangements with the same combinatorics. We then study the situation when the hypothesis does not hold. For equidimensional curves in \mathbb P3, the Hartshorne-Rao module from liaison theory measures the failure of an ideal to be Cohen-Macaulay, degree by degree, and also determines the even liaison class of such a curve. We show that for any positive integer r there is an arrangement \mathcal A for which R/ J fails to be Cohen-Macaulay in only one degree, and this failure is by r; we also give an analogous result for √(J). We draw consequences for the corresponding even liaison class of the curve defined by J or by √(J).