Some Algebraic Properties of the Schechtman–Varchenko Bilinear Forms

We examine a bilinear form associated with a real arrangement of hyperplanes introduced in [Schechtman and Varchenko 1991]. 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 arrangement. 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. 1. The Varchenko B Matrices Let A = {H1, . . . , Hl} be an arrangement of hyperplanes in R and let r(A) = {R1, . . . , Rm} denote the set of regions in the complement of the union of A. Let L(A) denote the collection of intersections of hyperplanes in A. Included in L(A) is R, which we think of as the intersection of the empty set of hyperplanes. We order the elements of L(A) by reverse inclusion thus making it into a poset. It is well known that this poset is a meet semilattice and is a geometric lattice if the arrangement is central. We will abbreviate L(A) to L when the arrangement is clear. For regions S, T ∈ r(A), define H(S, T ) to be the set of hyperplanes in A which separate S from T . Varchenko [1993] defines a matrix B = B(A) with rows and columns indexed by the regions in r(A) by saying that the S, T entry in B is ∏ H∈H(S,T ) aH , where aH is an indeterminate assigned to the hyperplane H. We will call B = B(A) the Varchenko matrix of the arrangement A. Example 1.1. As a starting example, let F = {H0, H1, H2} be the arrangement in R where Hj is the line y = (−1)x for j = 0, 1 and where H2 is the line y = 1. Note that r(F ) consists of 7 regions. Let these regions be numbered R1, . . . , R7, This research was partially supported by the National Science Foundation.