Related papers: On Left regular bands and real Conic-Line arrangem…
Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set 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…
In this paper we study plus-one generated arrangements of conics and lines in the complex projective plane with simple singularities. We provide several degree-wise classification results that allow us to construct explicit examples of such…
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular…
We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…
We define MC left regular bands and study their adjacency graphs. We prove that for thin MC left regular bands, the adjacency graph is particularly nice and is represented by edge labeled graphs where every simple cycle has an even number…
We list all the possible fundamental groups of the complements of real conic-line arrangements with two conics which are tangent to each other at two points, with up to two additional lines. For the computations we use the topological local…
A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is…
We present two results on the relation between the class of right regular bands (RRBs) and their underlying *associative posets*. The first one is a construction of a left adjoint to the forgetful functor that takes an RRB $(P,\cdot)$ to…
A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. If g is a Lie algebra with such a structure then its complexification has a hypercomplex structure. It is…
The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic…
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid…
The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain class of conic-line…
In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of $21$ lines in the complex projective plane.
We propose an approach to study logarithmic sheaves T(-log A) associated with a hyperplane arrangements A on the projective space, based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line…
In this paper, we study combinatorial aspects of reduced plane curves known as $\mathscr{M}$-curves. This notation is a natural generalization of maximizing plane curves which are well-known in the theory of algebraic surfaces. We focus…
We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable…
A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if…
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line…
Let $\mathcal C :f=0$ be a curve arrangement in the complex projective plane. If $\mathcal C$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains an explicit syzygy among the partial…