Some algebraic properties of the Schechtman-Varchenko bilinear forms
Phil Hanlon and Graham Denham
We examine a bilinear form associated with a real arrangement of hyperplanes introduced by Schechtman and Varchenko. Our main objective is to show that the linear algebraic properties of this bilinear form are related to the combinatorics and topology of the hyperplane arrangmenent. We will survey results and state a number of open problems which relate the determinant, cokernel structure and Smith normal form of the bilinear form to combinatorial and topological invariants of the arrangement including the characteristic polynomial, combinatorial structure of the intersection lattice and homology of the Milnor fibre.