Jumping coefficients and spectrum of a hyperplane arrangement

Abstract

Abstract. In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement as conjectured by Mustaţǎ. For this we proved a similar assertion on the spectrum. After this first proof was written, the first named author found a more conceptual proof using the Hirzebruch-Riemann-Roch theorem where the assertion on the jumping numbers was proved without reducing to that for the spectrum. In this paper we improve these methods and show that the jumping numbers and the spectrum are calculable in low dimensions without using a computer. In the reduced case we show that these depend only on fewer combinatorial data, and give completely explicit combinatorial formulas for the jumping coefficients and (part of) the spectrum in the case the ambient dimension is 3 or 4. We also give an analogue of Mustaţǎ's formula for the spectrum.

A computational approach to Milnor fiber cohomology

“…Note that the mixed Hodge structure on H 2 (F, C) =1 is not pure in general. For a line arrangement, one can use the formulas for the spectrum given in [6] to study the interplay between monodromy and Hodge structure on H 2 (F, C) =1 .…”

The Monodromy Conjecture for hyperplane arrangements

“…Corollary 1 in [1] implies that 3/d is a jumping number for the multiplier ideals of g if and only if 2d/3<m≤d…”

“…We will state a stronger result, for moderate arrangements, in Theorem 3.6.Remark 3.4 Even in the case of hyperplane arrangements defined by a reduced polynomial g, in general it is not the case that one can prove Conjecture 1.2 by showing that m/d is a jumping number for the multiplier ideals of g. For example, if g = x y(x − y)(x + y)(x + z), then 3/5 is not a jumping number (one can use, for example, Corollary 1 in[1]). However, this arrangement is decomposable, so Conjecture 1.1-(b) holds by Remark 3.2.…”

“…To state it, we fix some notation. For x ∈ f −1 (0), let M f,x denote the Milnor fiber of f at x, which is defined as the intersection of f −1 (t) with a small ball of radius around x (0 < t 1). It is well-known, by results of Malgrange [12] and Kashiwara [11], that all the roots of the b-function are negative rational numbers.…”

The Poincaré–Deligne Polynomial of Milnor Fibers of Triple Point Line Arrangements is Combinatorially Determined

“…On the positive side, it was shown by N. Budur and M. Saito in [2] that the spectrum Sp(A) of A, whose definition is also recalled in the next section, is combinatorially determined.…”