Bicolor-eliminable graphs and free multiplicities on the braid arrangement

We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley’s classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs. 0 Introduction Let V = V l be an l-dimensional vector space over a field K of characteristic zero, {x1, . . . , xl} a basis for the dual vector space V ∗ and S := Sym(V ) ≃ K[x1, . . . , xl]. Let DerK(S) denote the S-module of K-linear derivations of S, i.e., DerK(S) = ⊕l i=1 S · ∂xi . A non-zero element θ = ∑l i=1 fi∂xi ∈ DerK(S) is homogeneous of degree p if fi is zero or homogeneous of degree p for each i. A hyperplane arrangement A (or simply an arrangement) is a finite collection of affine hyperplanes in V . If each hyperplane in A contains the origin, we say that A is central. In this article we assume that all arrangements are central unless otherwise specified. A multiplicity m on an arrangement A is a map m : A → Z≥0 and a pair (A, m) is called a multiarrangement. Let |m| denote the sum of the multiplicities ∑ H∈A m(H). When m ≡ 1, (A, m) is the same as the hyperplane arrangement A and sometimes called a simple ∗Supported by 21st Century COE Program “Mathematics of Nonlinear Structures via Singularities” Hokkaido University.