Related papers: Free Arrangements over Finite Field
A toric hyperplane is the preimage of a point $x \in S^1$ of a continuous surjective group homomorphism $\theta: \mathbb{T}^n \to S^1$. A finite hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the…
We construct counterexamples to Yoshinaga's conjecture that every free arrangement is either inductively free or rigid in characteristic zero. The smallest example has $13$ hyperplanes, its intersection lattice has a one dimensional moduli…
In the present note we study determinantal arrangements constructed with use of the $3$-minors of a $3 \times 5$ generic matrix of indeterminates. In particular, we show that certain naturally constructed hypersurface arrangements in…
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then…
In this article, we study freeness of hyperplane arrangements. One of the most investigated arrangement is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal and Dirac…
A topological hyperplane is a subspace of R^n (or a homeomorph of it) that is topologically equivalent to an ordinary straight hyperplane. An arrangement of topological hyperplanes in R^n is a finite set H such that k topological…
Saito's criterion is a foundational result that algebraically characterizes free hyperplane arrangements via the determinant of a square matrix of logarithmic derivations. It is natural to ask whether this criterion can be generalized to…
The characteristic polynomial plays an important role in study of hyperplane arrangements. There are several refinements of the characteristic polynomial. One of them is the coboundary polynomial defined by Crapo. Another refinement is the…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…
We propose an approach to study logarithmic sheaves T(-log A) associated with a hyperplane arrangements A on the projective space, based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line…
Given a real finite hyperplane arrangement A and a point p not on any of the hyperplanes, we define an arrangement vo(A,p), called the *valid order arrangement*, whose regions correspond to the different orders in which a line through p can…
The addition-deletion theorems for hyperplane arrangements, which were originally shown in [H. Terao, Arrangements of hyperplanes and their freeness I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide useful ways to…
We consider arrangements of n hyperplanes of codimension one in a real projective space of dimension d. Let us denote by F the maximal possible number f of connected components of the complement in the projective space of dimension d to the…
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…
A hyperplane arrangement $\cA$ is said to be free if the corresponding Jacobian ideal $J_\cA$ is Cohen-Macaulay. If $\cA$ is free then $J_\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not…
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
Let A be an arrangement of complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism from a finitely generated free group to the pure braid group. Using the Gassner representation of…
We give lower bounds for the degree of multiplicative combinations of iterates of rational functions (with certain exceptions) over a general field, establishing the multiplicative independence of said iterates. This leads to a…
We show that for every lattice packing of $n$-dimensional spheres there exists an $(n/\log_2(n))$-dimensional affine plane which does not meet any of the spheres in their interior, provided $n$ is large enough. Such an affine plane is…
This paper studies the algebraic structure of a new class of hyperplane arrangement $A$ obtained by deleting two hyperplanes from a free arrangement. We provide information on the minimal free resolutions of the logarithmic derivation…