Related papers: Some analogs of Zariski's Theorem on nodal line ar…

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…

A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ and points $X$ satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac-Moody…

We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…

We study the topology of the boundary manifold of a line arrangement in CP^2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Delta(G), and more generally, the twisted Alexander…

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no…

We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…

In this paper we show a Zariski pair of sextics which is not a degeneration of the original example given by Zariski. This is the first example of this kind known. The two curves of the pair have a trivial Alexander polynomial. The…

We introduce an algorithm that exploits a combinatorial symmetry of an arrangement in order to produce a geometric reflection between two disconnected components of its moduli space. We apply this method to disqualify three real examples…

In a previous work, the third named author found a combinatorics of line arrangements whose realizations live in the cyclotomic group of the fifth roots of unity and such that their non-complex-conjugate embedding are not topologically…

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology of a line arrangement which generalizes the $\mathcal{I}$-invariant introduced by Artal, Florens and the author. This new invariant is called the loop…

We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and…

We construct two combinatorially equivalent line arrangements in the complex projective plane such that the fundamental groups of their complements are not isomorphic. The proof uses a new invariant of the fundamental group of the…

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by…

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to…

The boundary of the Milnor fiber associated with a complex line arrangement is a three dimensional plumbed manifold, and it is a combinatorial invariant. We prove the reverse implication, which was conjectured N\'emethi and Szil\'ard. That…

In this work we consider holomorphic foliations of degree two on the projective plane $\mathbb{P}^2$ having an invariant line. In a suitable choice of affine coordinates these foliations are induced by a quadratic vector field over the…

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. We show that for all even $g…

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…

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a…