Related papers: Arithmetic non-very generic arrangements

The discriminantal arrangement $\mathcal{B}(n,k,\mathcal{A})$ has been introduced by Manin and Schectman in 1989 and it consists of all non-generic translates of a generic arrangement $\mathcal{A}$ of n hyperplanes in a $k$-dimensional…

In 1985 Crapo introduced in \cite{Crapo} a new mathematical object that he called $\textit{geometry of circuits}$. Four years later, in 1989, Manin and Schechtman defined in \cite{MS} the same object and called it $\textit{discriminantal…

The discriminantal arrangement is the space of configurations of $n$ hyperplanes in generic position in a $k$ dimensional space (see \cite{MS}). Differently from the case $k=1$ in which it corresponds to the well known braid arrangement,…

In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of 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…

Manin and Schechtman introduced a family of arrangements of hyperplanes generalizing classical braid arrangements, which they called the $\textit{discriminantal arrangements}$. Athanasiadis proved a conjecture by Bayer and Brandt providing…

Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is…

In this article we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated…

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection…

In this paper, we present a combinatorial characterization of the hyperplanes associated with non-singular hermitian varieties ${H}\left(s, q^2\right)$ in the projective space $\mathrm{PG}\left(s,q^2\right)$ where $s\geq3$ and $q>2$. By…

In 1989 Manin and Schechtman defined the discriminantal arrangement $\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement $\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. An equivalent notion was already introduced…

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of…

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…

An algorithm is demonstrated that finds an ordinary intersection in an arrangement of $n$ lines in $\mathbb{R}^2$, not all parallel and not all passing through a common point, in time $O(n \log{n})$. The algorithm is then extended to find…

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic or topological, is of a combinatorial nature (that is determined by the intersection lattice) are abundant in the literature. To tackle…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mn\"ev's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it…

We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3, n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^3$ with associated discriminantal arrangement having intersections of multiplicity…

Let $B$ be an arrangement of linear complex hyperplanes in $C^d$. Then a classical result by Orlik \& Solomon asserts that the cohomology algebra of the complement can be constructed from the combinatorial data that are given by the…

For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair…