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Benoît Guerville-Ballé

Publications and source records attributed to Benoît Guerville-Ballé.

18 recordsLinked to original sources
arXiv2026

LA-MOKA: A Combinatorial Discretization Algorithm for the Braid Monodromy of Line Arrangements

Computing braid monodromy is a key tool for studying the topology of algebraic curves in the complex projective plane $\mathbb{P}^2$. We present an algorithm, named LA-MOKA, to compute this invariant in the specific case of complex line arrangements. To safely manage the problem of floating-point approximations when working over number fields, we translate the continuous geometry into a discrete combinatorial structure. We establish explicit conditions on this discretization that guarantee the topological correctness of the computed braid monodromy.

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arXiv2024

On the nonconnectedness of moduli spaces of arrangements, II: construction of nonarithmetic pairs

Constructing lattice isomorphic line arrangements that are not lattice isotopic is a complex yet fundamental task. In this paper, we focus on such pairs but which are not Galois conjugated, referred to as nonarithmetic pairs. Splitting polygons have been introduced by the author to facilitate the construction of lattice isomorphic arrangements that are not lattice isotopic. Exploiting this structure, we develop two algorithms which produce nonarithmetic pairs: the first generates pairs over a number field, while the second yields pairs over the rationals. Moreover, explicit applications of these algorithms are presented, including one complex, one real, and one rational nonarithmetic pair.

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arXiv2024

Connectedness and combinatorial interplay in the moduli space of line arrangements

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 connectedness 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.

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arXiv2021

On the non-connectivity of moduli spaces of arrangements: the splitting-polygon structure

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.

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arXiv2020

The loop linking number of line arrangements

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 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.

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arXiv2020

Ineffectiveness of homotopical invariants on Nakanishi's 4-move conjecture

A $4$-move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant of knots based on the fundamental group $π_1(S^3\setminus K)$ and preserved by $4$-moves is constant among the isotopy classes of knots.

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arXiv2019

A linking invariant for algebraic curves

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.

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arXiv2018

Configurations of points and topology of real line arrangements

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ 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 $\mathbb{RP}^2$, 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 $\mathbb{CP}^2$ (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 $\mathbb{Q}$ 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 three geometric characterizations of these components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.

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arXiv2018

Fundamental groups of real arrangements and torsion in the lower central series quotients

By using computer assistance, 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 for a problem of Falk and Randell. 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.

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arXiv2018

Topology and homotopy of lattice isomorphic arrangements

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.

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arXiv2017

Non-homotopicity of the linking set of algebraic plane curves

The linking set is an invariant of algebraic plane curves introduced by Meilhan and the first author. It has been successfully used to detect several examples of Zariski pairs, i.e. curves with the same combinatorics and different embedding in $\mathds{CP}^2$. Differentiating Shimada's $π_1$-equivalent Zariski pair by the linking set, we prove, in the present paper, that this invariant is not determined by the fundamental group of the curve.

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arXiv2016

On the topology of arrangements of a cubic and its inflectional tangents

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.

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arXiv2016

Multiplicativity of the $\mathcal{I}$-invariant and topology of glued arrangements

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, 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.

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arXiv2015

On complex line arrangements and their boundary manifolds

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.

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arXiv2015

An arithmetic Zariski 4-tuple of twelve lines

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 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 $\mathbb{CP}^2$.

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arXiv2015

Combinatorics of line arrangements and polynomial vector fields

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ 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 $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.

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arXiv2015

Combinatorics of line arrangements and dynamics of polynomial vector fields

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$--derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We 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 $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.

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