Related papers: Locally Heavy Hyperplanes in Multiarrangements

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given. It…

We consider an arrangement $\A$ of $n$ hyperplanes in $\R^d$ and the zone $\Z$ in $\A$ of the boundary of an arbitrary convex set in $\R^d$ in such an arrangement. We show that, whereas the combinatorial complexity of $\Z$ is known only to…

If F is a master function corresponding to a hyperplane arrangement A and a collection of weights y, we investigate the relationship between the critical set of F, the variety defined by the vanishing of the one-form w = d log F, and the…

We prove that if $G$ is an abelian group and $H_1x_1,\dots,H_{k}x_k$ is an irredundant (minimal) cover of $G$ with cosets, then $$|G:\bigcap_{i=1}^{k}H_{i}|=2^{O(k)}.$$ This bound is the best possible up to the constant hidden in the…

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an $ E_3 $-algebra (respectively $ E_2 $-algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric…

The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n…

Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a…

There are two invariants associated to any line arrangement: the freeness defect $\nu(C)$ and an upper bound for it, denoted by $\nu'(C)$, coming from a recent result by Uli Walther. We show that $\nu'(C)$ is combinatorially determined, at…

This paper is based on the observation that dimension of weight spaces of multi-variable Weyl modules depends polynomially on the highest weight (Conjecture 1). We support this conjecture by various explicit answers for up to three variable…

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk…

In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to…

We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…

In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement…

We study Pythagorean hyperplane arrangements, originally defined by Zaslavsky. In this first part of a series on such arrangements, we introduce a new notion of genericity for such arrangements. Using this notion we construct an auxiliary…

A conjecture of Denef-Jacobs-Veys relates motivic principal value integrals of multivalued rational top-forms with cohomology support loci of rank one local systems. We give a stronger positive answer to this conjecture for hyperplane…

We study the rank-three lifting problem for incidence matrices of finite projective planes through residue-level determinant constraints invisible to tropical valuations alone. In residue characteristic $\neq 3$, any rank-$\le 3$ lift of…

We use covering space theory and the fundamental group of complements of complexified-real line arrangements to explore the associated Milnor fiber. This work yields a combinatorially determined upper bound on the rank of the first homology…

We study the notion of a nice partition or factorization of a hyperplane arrangement due to Terao from the early 1990s. The principal aim of this note is an analogue of Terao's celebrated addition-deletion theorem for free arrangements for…

There are two restriction maps of the logarithmic modules of plane arrangements in a three dimensional vector space. One is the Euler restriction and the other is the Ziegler restriction. The dimension of the cokernel of the Ziegler…

Let $\A$ be a free arrangement of $d$ lines in the complex projective plane, with exponents $d_1\leq d_2$. Let $m$ be the maximal multiplicity of points in $\A$. In this note, we describe first the simple cases $d_1 \leq m$. Then we study…