Efficient computation of resonance varieties via Grassmannians

Associated to the cohomology ring A of the complement X (A) of a hyperplane arrangement A in C^ℓ are the resonance varieties R^k (A). The most studied of these is R¹ (A), which is the union of the tangent cones at 1 to the characteristic varieties of π₁ (X (A)). R¹ (A) may be described in terms of Fitting ideals, or as the locus where a certain E x t module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R¹ (A) as the locus of decomposable two-tensors in the Orlik-Solomon ideal of A leads to a description of R¹ (A) as the intersection of a Grassmannian with a linear space, determined by the quadratic generators of the Orlik-Solomon ideal. This method is much faster than previous alternatives.