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This paper aims to undertake an exploration of the behavior of the moduli space of line arrangements while establishing its combinatorial interplay with the incidence structure of the arrangement. In the first part, we investigate combinatorial classes of arrangements whose moduli space is connected. We unify the classes of simple and inductively connected arrangements appearing in the literature. Then, we introduce the notion of arrangements with a rigid pencil form. It ensures the connectivity of the moduli space and is less restrictive that the class of C_3 arrangements of simple type. In the last part, we obtain a combinatorial upper bound on the number of connected components of the moduli space. Then, we exhibit examples with an arbitrarily large number of connected components for which this upper bound is sharp.

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 such questions and provide a negative answer, one of the most effective methods is to produce a counterexample. To this end, it is essential to know how to construct arrangements that are lattice-equivalent. The more different they are, the more efficient it will be.
In this paper, we present a method to construct arrangements of complex projective lines that are lattice-equivalent but lie in distinct connected components of their moduli space. To illustrate the efficiency of the method, we apply it to reconstruct all the classical examples of arrangements with disconnected moduli spaces: MacLane, Falk-Sturmfels, Nasir-Yoshinaga and Rybnikov. Moreover, we employ this method to produce novel examples of arrangements of eleven lines whose moduli spaces are formed by four connected components.

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology of a line arrangement which generalizes the I-invariant introduced by Artal, Florens and the author. This new invariant is called the loop-linking number in the present paper. We refine the result of Cadegan-Schlieper by proving that the loop-linking number is an invariant of the homeomorphism type of the arrangement complement.
We give two effective methods to compute this invariant, both are based on the braid monodromy. As an application, we detect an arithmetic Zariski pair of arrangements with 11 lines whose coefficients are in the 5th cyclotomic field. Furthermore, we also prove that the fundamental groups of their complements are not isomorphic; it is the Zariski pair with the fewest number of lines which have this property. We also detect an arithmetic Zariski triple with 12 lines whose complements have non-isomorphic fundamental groups. In the appendix, we give 28 similar arithmetic Zariski pairs detected using the loop-linking number.
To conclude this paper, we give a multiplicativity theorem for the union of arrangements. This first allows us to prove that the complements of Rybnikov's arrangements are not homeomorphic, and then leads us to a generalization of Rybnikov's result. Lastly, we use it to prove the existence of homotopy-equivalent lattice-isomorphic arrangements which have non-homeomorphic complements.

We prove the existence of lattice isomorphic line arrangements having π_1-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is formed by real-complexified arrangements while the second is not.

We prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example to a Falk-Randell Problem. We also deduce that the torsion of the lower central series quotients is not combinatorially determined, which gives a negative answer to a question of Suciu.

A central question in the study of line arrangements in the complex projective plane is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane, dealing only with geometrical properties.
Using this dual point of view, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in the complex projective plane (i.e. Zariski pairs), which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over the rational and containing only double and triple points. For each one of them, we can derive degenerations, containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.
We explicitly compute the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain two geometric characterizations of these components: the existence of a conic tangent to six lines as well as the collinearity of three specific triple points.

A k-Artal arrangement is a reducible algebraic curve composed of a smooth cubic and k inflectional tangents. By studying the topological properties of their subarrangements, we prove that for k=3,4,5,6, there exist Zariski pairs of k-Artal arrangements. These Zariki pairs can be distinguished in a geometric way by the number of collinear triples in the set of singular points contained in the cubic.

The splitting numbers and the linking set are two invariants of the topology of algebraic plane curves. They were introduced by the second authors for the first and by JB-Meilhan and the first author for the second one. Although they come from different areas of the mathematics -algebraic geometry for the splitting numbers and geometric topology for the linking set- we prove in this paper that they are equivalent in particular cases. This allows us to deduce that the linking set is not determined by the fundamental group of the complement of a curve.

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 the first author with Artal and Florens. We give two practical tools for computing this invariant, using a modification of the usual braid monodromy or using the connected numbers introduced by Shirane. As an application, we show that this invariant distinguishes several Zariski pairs, i.e. pairs of curves having same combinatorics, yet different topologies. The former is the well known Zariski pair found by Artal, composed of a smooth cubic with 3 tangent lines at its inflexion points. The latter is formed by a smooth quartic and 3 bitangents.

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 equivalent in the sense that they are not embedded in the same way in the complex projective plane. That work does not imply that the complements of the arrangements are not homeomorphic. In this work we prove that the fundamental groups of the complements are not isomorphic. It provides the first example of a pair of Galois-conjugate plane curves such that the fundamental groups of their complements are not isomorphic (despite the fact that they have isomorphic profinite completions).

The I-invariant was first introduced in [4]. Inspired by idea of G. Rybnikov in [9], we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnikov arrangements form an ordered Zariski pairs (i.e. two arrangements with the same combinatorial information and different ordered topologies). Finally, we extend this method to a particular family of arrangements and thus we obtain a method to construct new examples of Zariski pairs.

Let A be a real line arrangement and D(A) the module of A-derivations. We first give a dynamical interpretation of D(A) as the set of polynomial vector fields which posses A as an invariant set. We first characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in D(A) is not determined by the combinatorics of A.

Using the invariant developed in [AFG], 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 orientation-preserving homeomorphism between them. Furthermore, some couples of arrangements among this 4-tuplet form new arithmetic Zariski pairs, i.e. a couple of arrangements with the same combinatorial information but with different embedding in CP^2.

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 fundamental group of their complements. It is derived from the peripheral structure on the group induced by the inclusion map of the boundary of a tubular neigborhood in the exterior of the arrangement. By similarity with knot theory, it can be viewed as an analogue of linking numbers. This is an orientation-preserving invariant for ordered arrangements. We give an explicit method to compute the invariant from the equations of the arrangement, by using wiring diagrams introduced by Arvola, that encode the braid monodromy. Moreover, this invariant is a crucial ingredient to compute the depth of a character satisfying some resonant conditions, and complete the existent methods by Libgober and the first author. Finally, we compute the invariant for extended MacLane arrangements with an additional line and observe that it takes different values for the deformation classes.

Let A be a line arrangement in the complex projective plane CP2. We define and describe the inclusion map of the boundary manifold -the boundary of a close regular neighborhood of A- in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computations provide a new minimal presentation of the fundamental group of the complement, generalizing Randell's presentation.