Related papers: Simplicial arrangements with up to 27 lines
A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb P^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case…
We consider the problem of stretching pseudolines in a planar straight-line drawing to straight lines while preserving the straightness and the combinatorial embedding of the drawing. We answer open questions by Mchedlidze et al. by showing…
Given $n$ lines in general position in the plane, how many bounded triangular faces can the arrangement have? We construct a straight-line affine arrangement of $19$ lines satisfying the conditions of the iterative construction by…
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…
In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as $T_{1} \sqcup T_{2}$, such that each $T_{i}$ represents the lines of a copy of the Fano plane $PG(2, \mathbb{F}_{2})$. We generalize this observation by…
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…
In this, the last of three papers about $C_2$-equivariant complex quadrics, we complete the calculation of the equivariant ordinary cohomology of smooth symmetric quadrics in the cases where the fixed sets have more than two components.…
Two line arrangements in $\mathbb{CP}^2$ can have different topological properties even if they are combinatorially isomorphic. Results by Dan Cohen and Suciu and by Randell show that a reducible moduli space under complex conjugation is a…
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…
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…
We give restrictions on the weak combinatorics of line arrangements with singular points of odd multiplicity using topological arguments on locally-flat spheres in 4-manifolds. As a corollary, we show that there is no line arrangement…
In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we…
We study the topological complexity, in the sense of Smale, of three enumerative problems in algebraic geometry: finding the 27 lines on cubic surfaces, the 28 bitangents and the 24 inflection points on quartic curves. In particular, we…
We present a technique to produce arrangements of lines with nice properties. As an application, we construct $(22_4)$ and $(26_4)$ configurations of lines. Thus concerning the existence of geometric $(n_4)$ configurations, only the case…
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…
In this article an explicit method (relying on representation theory) to construct packings in Grassmannian space is presented. Infinite families of configurations having only one non-trivial set of principal angles are found using…
Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the…
The simple permutations in two permutation classes --- the 321-avoiding permutations and the skew-merged permutations --- are enumerated using a uniform method. In both cases, these enumerations were known implicitly, by working backwards…
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…
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…