In this note we determine the Bernstein-Sato polynomial bQ(s) of a generic central arrangement Q = ∏k i=1 Hi of hyperplanes. We establish a connection between the roots of bQ(s) and the degrees of the generators for the top cohomology of the corresponding Milnor fiber. This connection holds for all homogeneous polynomials. We also introduce certain subschemes of the arrangement determined by th...

Any finite set of lines in the real projective plane determines a cell complex; these complexes and their combinatorial properties have been a subject of study at least since 1826 [9]. More recently, Levi [6] considered a topological generalization of this notion, defined as follows: Consider a simple closed curve in RP2 which does not separate RP2; this is called a pseudoline. (It is clear tha...

We introduce a new combinatorial object, the double-permutation sequence, and use it to encode a family of mutually disjoint compact convex sets in the plane in a way that captures many of its combinatorial properties. We use this encoding to give a new proof of the Edelsbrunner-Sharir theorem that a collection of n compact convex sets in the plane cannot be met by straight lines in more than 2...

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.

We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this GaussManin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this Gauss-Manin connection for all arrangements.

To a set of n points in the plane, one can associate a graph that has less than n2 vertices and has the property that k-cliques in the graph correspond vertex sets of convex k-gons in the point set. We prove an upper bound of 2k−1 on the size of a planar point set for which the graph has chromatic number k, matching the bound conjectured by Szekeres for the clique number. Constructions of Erdős...

An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the Gauss-Manin connection for the moduli space of arrangements of fixed combinatorial type in the cohomology of the complement with coefficients in the local system deter...

We define the decision problem data arrangement, which involves arranging the vertices of a graph G at the leaves of a d-ary tree so that a weighted sum of the distances between pairs of vertices measured with respect to the tree topology is at most a given value. We show that data arrangement is strongly NP-complete for any fixed d ≥ 2 and explain the connection between data arrangement and ar...