Personal Information
About me
Since May 2025, I've been researcher at Sassari University. Before, I had positions in RIMS (Kyoto University), Kyushu University, Polish Academy of Sciences, São Paulo University, Tokyo Gakugei University, Fourier Institute and University of Pau.
Outside academia, I like outdoor activities. I regularly practice trail running, bicycle/bikepacking and hiking.
Research Interests
My research focuses on the study of line arrangements and algebraic plane curves in the complex projective plane. I'm interested in the interaction between their geometry, their topology and their local singularities. My passion is to discover Zariski pairs with unexpected differences, i.e., couple of arrangements with the same combinatorics yet different topologies.
Keywords: Geometric Topology, Line arrangements, Algebraic plane curves, Zariski pairs, Moduli spaces, Algebraic computations (Sagemath).
Contact Information
Address:
Matematica, Università di Sassari,
via Vienna 2, 07100 Sassari, Italy
Email: benoit.guerville-balle(at)math.cnrs.fr
Email: baguerville(at)uniss.it
Publications
Preprints
18
LA-MOKA: A combinatorial discretization algorithm for the braid monodromy of line arrangements
Preprint: arXiv:2607.29333 [math.AG]
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.
Publications
17
Zariski pairs of conic-line arrangements with a unique conic
To be published in Proc. Am. Math. Soc.
arXiv: 2410.04969 [math.AG]
DOI: 10.1090/proc/17449
In this note, we present two pairs of conic-line arrangements admitting a unique conic and that form Zariski pairs, both of degree 9. Their topologies are distinguished using the connected numbers.
16
On the nonconnectedness of moduli spaces of arrangements, II: construction of nonarithmetic pairs
To be published in Kyoto J. Math.
arXiv: 2409.18022 [math.AG]
DOI: 10.1215/21562261-2025-0024
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.
15
Connectedness and combinatorial interplay in the moduli space of line arrangements
Contemporary Mathematics, Volume 843, (2026), Proceedings Volume: 115AM. Algebraic and topological interplay of algebraic varieties,
arXiv: 2309.00322 [math.AG]
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.
14
On the nonconnectedness of moduli spaces of arrangements: the splitting polygon structure
Kyoto J. Math. 65, No. 1, 153-170 (2025)
arXiv: 2111.00399 [math.AG]
DOI: 10.1215/21562261-2024-0011
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.
13
The loop-linking number of line arrangements
Math. Z. 301, No. 2, 1821-1850 (2022)
arXiv: 2004.03550 [math.AG]
DOI: 10.1007/s00209-021-02953-x
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.
12
A linking invariant for algebraic curves
Enseign. Math. (2) 66, No. 1-2, 63-81 (2020)
arXiv: 1602.04916 [math.AG]
DOI: 10.4171/LEM/66-1/2-4
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.
11
Fundamental groups of real arrangements and torsion in the lower central series quotients
Exp. Math. 29, No. 1, 28-35 (2020)
arXiv: 1704.04152 [math.AG]
DOI: 10.1080/10586458.2018.1428131
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.
10
Topology and homotopy of lattice isomorphic arrangements
Proc. Am. Math. Soc. 148, No. 5, 2193-2200 (2020)
arXiv: 1801.02682 [math.AG]
DOI: 10.1090/proc/14878
We prove the existence of lattice isomorphic line arrangements having π1-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.
9
Configurations of points and topology of real line arrangements
Math. Ann. 374, No. 1-2, 1-35 (2019)
arXiv: 1702.00922 [math.AG]
DOI: 10.1007/s00208-018-1673-0
A central question in the study of line arrangements in the complex projective plane 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 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 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 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.
8
Multiplicativity of the I-invariant and topology of glued arrangements
J. Math. Soc. Japan 70, No. 1, 215-227 (2018)
arXiv: 1506.08227 [math.AG]
The I-invariant 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.
7
A topological invariant of line arrangements
Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 3, 949-968 (2017)
arXiv: 1407.3387 [math.AG]
DOI: 10.2422/2036-2145.201509_014
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.
6
Non-homotopicity of the linking set of algebraic plane curves
J. Knot Theory Ramifications 26, No. 13, Article ID 1750089, 13 p. (2017)
arXiv: 1607.04951 [math.AG]
DOI: 10.1142/S0218216517500894
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 $\mathbb{CP}^2$. Differentiating Shimada's $\pi_1$-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.
5
On the topology of arrangements of a cubic and its inflectional tangents
Proc. Japan Acad., Ser. A 93, No. 6, 50-53 (2017)
arXiv: 1607.07618 [math.AG]
DOI: 10.3792/pjaa.93.50
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.
4
An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group
Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 111, No. 2, 377-402 (2017)
arXiv: 1507.00190 [math.AG]
DOI: 10.1007/s13398-016-0298-y
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).
3
On the minimal degree of logarithmic vector fields of line arrangements
Monografías Matemáticas "García de Galdeano" 40, 61-66 (2016)
arXiv: 1412.0137 [math.AG]
Let A be a real line arrangement and D(A) the module of A-derivations. First, we give a dynamical interpretation of D(A) as the set of polynomial vector fields which posses 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 D(A) is not determined by the combinatorics of A.
2
An arithmetic Zariski 4-tuple of twelve lines
Geom. Topol. 20, No. 1, 537-553 (2016)
arXiv: 1411.2300 [math.AG]
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 CP^2.
1
On complex line arrangements and their boundary manifolds
Math. Proc. Camb. Philos. Soc. 159, No. 2, 189-205 (2015)
arXiv: 1305.5645 [math.AG]
DOI: 10.1017/S0305004115000262
Let A be a line arrangement in the complex projective plane CP^2. 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.
Theses
PhD Thesis:
"Topological invariants of line arrangements" (December 2013)
HDR Thesis:
"A stroll through the topology of line arrangements and their moduli spaces" (December 2023)
Other
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