Related papers: Triangular arrangements on the projective plane
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…
In the present paper, we study conic-line arrangements having nodes, tacnodes, and ordinary triple points as singularities. We provide combinatorial constraints on such arrangements and we give the complete classification of free…
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…
We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have…
We show that there are only finitely many combinatorial types of free real line arrangements with only double, triple and quadruple intersection points, and we enlist all admissible weak-combinatorics of them. Then we classify all real…
A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been…
The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…
We study the geometry of $\mathcal{Q}$-conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such $\mathcal{Q}$-conic…
In the present note we study combinatorial and algebraic properties of cubic-line arrangements in the complex projective plane admitting nodes, ordinary triple and $A_{5}$ singular points. We deliver a Hirzebruch-type inequality for such…
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the…
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…
In this series of three articles, we give an exposition of various results and open problems in three areas of algebraic and geometric combinatorics: totally non-negative matrices, representations of the symmetric group, and hyperplane…
In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been…
In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number…
We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples…
The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining…
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…
Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets:…
Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise…
We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12…