Geometry and combinatorics of resonant weights

Let A be an arrangement of n hyperplanes in ℂ^ℓ. Let k be a field and A=⊕_p=0 ^ℓ A ^p the Orlik-Solomon algebra of A over k. The p ^th resonance variety of A over k is the set R^p (A, k) of one-forms a∈A ¹ annihilated by some b∈A ^p\(a). This notion arises naturally from consideration of cohomology of the rank-one local systems generated by single-valued branches of A-master functions Φ_a. For the most part we focus on the case p=1. We will describe the features of R ¹ (A, k) for k = C and also for fields of positive characteristic, and their connections with other phenomena. We derive simple necessary and sufficient conditions for an element a to lie in R ¹ (A, k), and consequently obtain a precise description of R ¹ (A, k) as a ruled variety. We sketch the description of components of R ¹ (A, C) in terms of multinets, and the related Ceva-type pencils of plane curves. We present examples over fields of positive characteristic showing that the ruling may be quite nontrivial. In particular, R ¹ (A, k) need not be a union of linear varieties, in contrast to the characteristic-zero case. We also give an example for which components of R ¹ (A, k) do not intersect trivially. We discuss the current state of the classification problem for Orlik-Solomon algebras, and the utility of resonance varieties in this context. Finally we sketch a relationship between one-forms a ∈ R ^p (A, k) and the critical loci of the corresponding master functions Φ_a. For p=1 we obtain a precise connection using the associated multinet and Ceva-type pencil.