Abstract. We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M , in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce a q-deformation of the Möbius algebra of M , and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples.

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition-deletion theorems to various arrangements including supersolvable arrangements and the Coxeter arrangement of type A3 to construct free and non-free multiarrangements. IntroductionLet A be a hyperplane arrangement, or simply an arrangement. In other words, A is a finite collection of hyperplanes in an ℓ-dimensional vector space V over a field K. A multiarrangement, which was introduced by Ziegler in [Z], is a pair (A, m) consisting of a hyperplane arrangement A and a multiplicity m : A → Z >0 . Define |m| = H∈A m(H). A multiarrangement (A, m) such that m(H) = 1 for all H ∈ A is just a hyperplane arrangement, and is sometimes called a simple arrangement.Let {x 1 , . . . , x ℓ } be a basis forWhen each H ∈ A contains the origin, we say that A is central. Throughout this article, assume that every arrangement is central. Let Der K (S) denote the set of K-linear derivations from S to itself. For each H ∈ A we choose a defining formIf D(A, m) is a free S-module we say that (A, m) is a free multiarrangement. In his groundbreaking paper [Z], Ziegler writes ". . . the theory of multiarrangements and their freeness is not yet in a satisfactory state. In particular, we do not know any addition/deletion theorem . . . ." It is exactly the subject of this article. Namely, we generalize the addition-deletion theorems for simple arrangements [T1] to multiarrangements in this article. Let (A, m) be a nonempty multiarrangement and ℓ ≥ 2. Fix a hyperplane H 0 ∈ A and let α 0 be a defining form for H 0 . To state the addition-deletion theorems for multiarrangements we need to define the deletion (A ′ , m ′ ) and the restriction (A ′′ , m ′′ ). First, we define the deletion as follows:Next we define the restriction (A ′′ , m ′′ ). Letwhich is an arrangement on H 0 . We, however, have more than one choice to define a multiplicity m ′′ . The definition of a suitable multiplicity m ′′ is crucial. The canonical definition is probablywhich is purely combinatorial and was used in [Y1, Y2, Z] effectively. In this article, however, in order to serve our purposes, we introduce a new multiplicity m * , called the Euler multiplicity, whose definition is algebraic rather than combinatorial.For X ∈ A ′′ define A X = {H ∈ A | X ⊂ H} and m X = m | AX .

Given a multiarrangement of hyperplanes we define a series by sums of the Hilbert series of the derivation modules of the multiarrangement. This series turns out to be a polynomial. Using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arragnement. The characteristic polynomial of an arrangement is a combinatorial invariant, but this generalized characteristic polynomial is not. However, when the multiarrangement is free, we are able to prove the factorization theorem for the characteristic polynomial. The main result is a formula that relates 'global' data to 'local' data of a multiarrangement given by the coefficients of the respective characteristic polynomials. This result gives a new necessary condition for a multiarrangement to be free. Consequently it provides a simple method to show that a given multiarrangement is not free.

Abstract. In this paper we consider an effective divisor on the complex projective line and associate with it the module D consisting of all the derivations θ such that θ(for every i, where I i is the ideal of p i . The module D is graded and free of rank 2; the degrees of its homogeneous basis, called the exponents, form an important invariant of the divisor. We prove that under certain conditions on (m i ) the exponents do not depend on {p i }. Our main result asserts that if these conditions do not hold for (m i ), then there exists a general position of n points for which the exponents do not change. We give an explicit formula for them. We also exhibit some examples of degeneration of the exponents, in particular, those where the degeneration is defined by the vanishing of certain Schur functions. As an application and motivation, we show that our results imply Terao's conjecture (concerning the combinatorial nature of the freeness of hyperplane arrangements) for certain new classes of arrangements of lines in the complex projective plane.