Efficient computation of resonance varieties via Grassmannians

Details Efficient computation of resonance varieties via Grassmannians Efficient computation of resonance varieties via Grassmannians

2008-08-14

arxiv.org/abs/0808.2027v2

Journal of Pure and Applied Algebra, 213 (2009), 1606-1611

Mathematics

Combinatorics

7 pages 2 figures, Macaulay2 code to implement v2: refs updated

Scientific paper

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.

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