Related papers: On Parallel Lines and Free Group

We prove that for any algebraic plane curve $C$ of degree at most $5$, the fundamental group $\pi_1(\mathbb CP^2\setminus C)$ is linear and virtually polyfree. As a consequence, we answer positively the open question on the residual…

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (resp. delete) a line to (resp. from) a free line arrangement, then the resulting line arrangement is either…

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…

We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.

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…

We produce a one-parameter family of hyperplane arrangements that are counterexamples to the conjecture of Saito that the complexified complement of a free arrangement is $K(\pi,1)$. These arrangements are the restriction of a one-parameter…

We prove that the automorphism group of a topological parallelism on real projective 3-space is compact. In a preceding article it was proved that at least the connected component of the identity is compact. The present proof does not…

The polynomial automorphisms of the affine plane over a field K form a group which has the structure of an amalgamated free product. This well-known algebraic structure can be used to determine some key results about the symmetry and…

The purpose of this paper is to give new examples of families of free singularities. We first show that a generic equidimensional subspace arrangement is free. Furthermore, we show that a product of two reduced Cohen-Macaulay subspaces is…

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 prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.

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…

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 construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary…

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.

We consider an homogeneous action of a finite group on a free linear category over a field in order to prove that the subcategory of invariants is still free. Moreover we show that the representation type is preserved when considering…

In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field which has global cyclicity can be represented isomorphically by a line arrangement with a given set of distinct slopes…

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated…

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…

Let A be an arrangement of complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism from a finitely generated free group to the pure braid group. Using the Gassner representation of…