Efficient computation of resonance varieties via Grassmannians

Associated to the cohomology ring AA of the complement X(A)X(A) of a hyperplane arrangement AA in CℓCℓ are the resonance varieties Rk(A)Rk(A). The most studied of these is R1(A)R1(A), which is the union of the tangent cones at 1 to the characteristic varieties of π1(X(A))π1(X(A)). R1(A)R1(A) may be described in terms of Fitting ideals, or as the locus where a certain ExtExt module is supported. Both these descriptions give obvious algorithms for computation. In this note, we show that interpreting R1(A)R1(A) as the locus of decomposable two-tensors in the Orlik–Solomon ideal of AA leads to a description of R1(A)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.