Related papers: Fermat-type arrangements
In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide…
In this paper we present new results about arrangements of lines and osculating curves associated to the Fermat curves in the projective plane. We first consider the sextactic points on the Fermat curves and show that they are distributed…
A hyperplane arrangement in $\mathbb{R}^n$ is a finite collection of affine hyperplanes. Counting regions of hyperplane arrangements is an active research direction in enumerative combinatorics. In this paper, we consider the arrangement…
We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We…
This paper studies \emph{Dirichlet arrangements}, a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize descriptions of combinatorial features of graphic…
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of…
We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link…
We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical…
The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb…
Some of the most classically relevant Hyperplane arrangements are the Braid Arrangements $B_n$ and their associated compliment spaces $\mathcal{F}_n$. In their recent work, Tsilevich, Vershik, and Yuzvinsky construct what they refer to as…
This is a short note on various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove…
This chapter is divided into two parts. The first part is a survey of some recent results on nests and the orderability problem. The second part consists of results, partial results and open questions, all viewed in the light of nests. From…
In this paper, we begin by introducing a well-known geometry concept: the Fermat point in a triangle. Then, we generalize the problem and propose an iterative algorithm based on gradient descent to the weighted form in Lp space. We also…
This paper introduces a general modeling framework for a multi-type maximal covering location problem in which the position of facilities in different metric spaces are simultaneously decided to maximize the demand generated by a set of…
We introduce a new family of hyperplane arrangements inspired by the homogenized Linial arrangement (which was recently introduced by Hetyei), and show that the intersection lattices of these arrangements are isomorphic to the bond lattices…
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…
The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by…
Unexpected hypersurfaces are a brand name for some special linear systems. They were introduced around 2017 and are a field of intensive study since then. They attracted a lot of attention because of their close tights to various other…
The article studies a generalization of the classical Fermat-Torricelli problem to normed spaces of arbitrary finite dimension. Necessary and sufficient conditions for the uniqueness of the solution of the Fermat-Torricelli problem for any…
Applying Baaz's Generalization Method and a new technique to, respectively, proofs and denumerable simple graphs, diverse arithmetical patterns are observed. In particular, sufficient conditions for a number to be a divisor of a Fermat…