Related papers: On the containment problem and sporadic simplicial…
In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines…
In the present note we provide a complete classification of nearly free (and not free simultaneously) simplicial arrangements of $d\leq 27$ lines.
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 compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Gr\"unbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines,…
Since Dumnicki, Szemberg and Tutaj-Gasi\'nska gave in 2013 in [9] the first example of a set of points in the complex projective plane such that for its homogeneous ideal I the containment of the third symbolic power in the second ordinary…
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…
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…
In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a…
We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free…
The main purpose of the note is to show that Yoshinaga's arrangement of $18$ lines having $48$ triple and $9$ double intersection points leads to a new (short) series of non-containment examples for $I^{(3)} \subset I^{2}$, the question…
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…
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…
The purpose of this work is to collect in one place available information on line arrangements known in the literature as braid, monomial, Ceva or Fermat arrangement. They have been studied for a long time and appeared recently in…
The main purpose of the present paper is to provide a partial classification, performed with respect the weak-combinatorics, of free arrangements consisting of lines and one smooth conic with quasi-homogeneous ordinary singularities.
In this note we focus on combinatorial aspects of plus-one generated line arrangements. We provide combinatorial constraints on such arrangements and we construct a polynomial that decodes the plus-one generated property. We present new…
In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, the conjecture by Terao asserting that freeness depends only on combinatorics holds true. A long standing…
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…
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)…
In the category of free arrangements, inductively and recursively free arrangements are important. In particular, in the former, Terao's open problem asking whether freeness depends only on combinatorics is true. A long standing problem…
There are four non-isomorphic configurations of triples that can form a triangle in a $3$-uniform hypergraph. Forbidding different combinations of these four configurations, fifteen extremal problems can be defined, several of which already…