Arrangements and local systems

We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Orlik-Solomon algebra with the connection operator. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the Orlik-Solomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds for the local system Betti numbers in terms of those of the Orlik-Solomon algebra, recovering a result of Libgober and Yuzvinsky. For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in non-resonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.Comment: LaTeX, 14 page

Ranking patterns of the unfolding model and arrangements

In the unidimensional unfolding model, given m objects in general position there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. By changing these m objects, we can generate various ranking patterns. It is natural to ask how many ranking patterns can be generated and what is the probability of each ranking pattern when the objects are randomly chosen? These problems are studied by introducing a new type of arrangement called mid-hyperplane arrangement and by counting cells in its complement.Comment: 29 pages, 2 figure

Arrangements in unitary and orthogonal geometry over finite fields

AbstractLet V be an n-dimensional vector space over Fq. Let Φ be a Hermitian form with respect to an automorphism σ with σ2 = 1. If σ = 1 assume that q is odd. Let A be the arrangement of hyperplanes of V which are non-isotropic with respect to Φ, and let L be the intersection lattice of A. We prove that the characteristic polynomial of L has n − v roots 1, q,…, qn − v− 1 where v is the Witt index of Φ