2024/11/14 by Martin de Borbon, de Borbon, Martin, Dmitri Panov +1
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Point processes and geometric inequalities #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2411.09573
openalex publication_date 2024/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a hyperplane arrangement in \mathbbCPn. We define a quadratic form Q on ℝH that is entirely determined by the intersection poset of H. Using the Bogomolov-Gieseker inequality for parabolic bundles, we show that if a ∈ ℝH is such that the weighted arrangement (H, a) is stable, then Q(a) ≤ 0. As an application, we consider the symmetric case where all the weights are equal. The inequality Q(a, …, a) ≤ 0 gives a lower bound for the total sum of multiplicities of codimension 2 intersection subspaces of H. The lower bound is attained when every H ∈ H intersects all the other members of H ∖ \H\ along (1-2/(n+1))|H| + 1 codimension 2 subspaces; extending from n=2 to higher dimensions a condition found by Hirzebruch for line arrangements in the complex projective plane.