Related papers: On arrangements of hyperplanes from connected subg…
In this paper, we study the class of free multiarrangements of hyperplanes. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over the rationals.
A configuration of lattice vectors is supernormal if it contains a Hilbert basis for every cone spanned by a subset. We study such configurations from various perspectives, including triangulations, integer programming and Groebner bases.…
In 2024, Bellamy, Craw, Rayan, Schedler, and Weiss introduced a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties which we thus call hyperpolygonal arrangements…
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present…
Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the $\mathbb{F}_1$-structure (in the sense of torification) on the complement of a hyperplane…
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,…
Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential…
We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
Abundant second-order maximally conformally superintegrable Hamiltonian systems are re-examined, revealing their underlying natural Weyl structure and offering a clearer geometric context for the study of St\"ackel transformations (also…
This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as…
Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where…
We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property…
We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly…
We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian…
A graph drawn on the plane is called $1$-plane if each edge is crossed at most once by another edge. In this paper, we show that every $4$-connected $1$-plane graph has a connected spanning plane subgraph. We also show that there exist…
An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we…
In the present note we study some arrangements of inflectional lines, hyperosculating conics, and a nodal plane cubic that are free. Moreover, we study weak combinatorics of arrangements consisting of lines, conics, and elliptic curves…
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive…
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…