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