Efficient computation of resonance varieties via Grassmannians

TL;DR

This paper introduces a faster method for computing the first resonance variety of hyperplane arrangements by interpreting it as an intersection of a Grassmannian with a linear space, improving computational efficiency.

Contribution

The authors present a novel approach that describes R^1(A) as a Grassmannian intersection, significantly enhancing the speed of computation over existing methods.

Findings

The new method is substantially faster than previous algorithms.

Interpreting R^1(A) as a Grassmannian intersection simplifies calculations.

The approach leverages the structure of the Orlik-Solomon ideal.

Abstract

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.