Related papers: On line arrangements with odd multiplicities
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…
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…
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 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 a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be…
We estimate the number of lines on a non-K3 quartic surface. Such a surface with only isolated double point(s) contains at most twenty lines; this bound is attained by a unique configuration of lines and by a surface with a certain limited…
In this note, we show that real line arrangements of type at most one, admitting only intersection points of multiplicity at most five, satisfy certain boundedness properties. In particular, we prove that a free real arrangement of $d$…
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 propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities…
We classify moduli spaces of arrangements of 10 lines with quadruple points. We show that moduli spaces of arrangements of 10 lines with quadruple points may consist of more than 2 disconnected components, namely 3 or 4 distinct points. We…
Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a…
The complete classification of (3,3)-nets and of (3,4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of…
We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines. We…
We describe a new infinite family of line arrangements in the projective plane with only triple points singularities and recover previously known examples.
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…
In the present note we provide a partial classification of nearly free conic-line arrangements in the complex plane having nodes, tacnodes, and ordinary triple points. In this setting, our theoretical bound tells us that the degree of such…
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…
We prove the sharp upper bound of at most $52$ lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than $48$ lines on smooth complex quartics.
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…
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…