Benoît Guerville-Ballé

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 running, 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 difference, 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

Zariski pairs of conic-line arrangements with a unique conic

with Shinzo Bannai, Taketo Shirane

Accepted for publication in Proc. Am. Math. Soc.

Preprint: arXiv:2410.04969 [math.AG] (2024)

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.

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

Accepted for publication in Kyoto J. Math.

Preprint: arXiv:2409.18022 [math.AG] (2024)

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.

Connectedness and combinatorial interplay in the moduli space of line arrangements

with Juan Viu-Sos

Accepted for publication in Contemporary Mathematics Proceedings Volume: 115AM. Algebraic and topological interplay of algebraic varieties

Preprint: arXiv:2309.00322 [math.AG] (2023)

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.

Publications

On the nonconnectedness of moduli spaces of arrangements: the splitting polygon structure

Kyoto J. Math. 65, No. 1, 153-170 (2025)

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 (i.e., 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.

The loop-linking number of line arrangements

Math. Z. 301, No. 2, 1821-1850 (2022)

DOI: 10.1007/s00209-021-02953-x

We construct an invariant of the embedded topology of a line arrangement which generalizes the I-invariant. This new invariant is called the loop-linking number. We prove 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 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.

A linking invariant for algebraic curves

with Jean-Baptiste Meilhan

Enseign. Math. (2) 66, No. 1-2, 63-81 (2020)

DOI: 10.4171/LEM/66-1/2-4

We construct a topological invariant of algebraic plane curves, which is an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements. We give two practical tools for computing this invariant. As an application, we show that this invariant distinguishes several Zariski pairs.

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

with Enrique Artal Bartolo, Juan Viu-Sos

Exp. Math. 29, No. 1, 28-35 (2020)

DOI: 10.1080/10586458.2018.1428131

We study torsion elements in the lower central series quotients of fundamental groups of real line arrangement complements. Using computational methods, we provide examples and investigate the relationship between torsion and the combinatorial structure of arrangements.

Topology and homotopy of lattice isomorphic arrangements

Proc. Am. Math. Soc. 148, No. 5, 2193-2200 (2020)

DOI: 10.1090/proc/14878

We investigate when lattice isomorphic arrangements share the same topological and homotopy properties. We provide conditions under which lattice isomorphism implies that the complements are homotopy equivalent, and give counterexamples when these conditions fail.

Configurations of points and topology of real line arrangements

with Juan Viu-Sos

Math. Ann. 374, No. 1-2, 1-35 (2019)

DOI: 10.1007/s00208-018-1673-0

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, called the chamber weight. Using this dual setting, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings (i.e. Zariski pairs) which are distinguished by this invariant. We compute explicitly the moduli space of the combinatorics of one example, and prove that it has exactly two connected components.

Multiplicativity of the I-invariant and topology of glued arrangements

J. Math. Soc. Japan 70, No. 1, 215-227 (2018)

DOI: 10.2969/jmsj/07017515

The I-invariant was introduced in previous work with Artal and Florens. Inspired by ideas of 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 ordered Zariski pairs. We extend this method to a particular family of arrangements and thus obtain a method to construct new examples of Zariski pairs.

A topological invariant of line arrangements

with Enrique Artal Bartolo, Vincent Florens

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 3, 949-968 (2017)

DOI: 10.2422/2036-2145.201509_014

We introduce a new topological invariant for line arrangements, called the I-invariant, derived from linking numbers in the complement. This invariant is computed using the braid monodromy of the arrangement. We demonstrate its effectiveness by distinguishing several Zariski pairs that could not be separated by previously known invariants.

Non-homotopicity of the linking set of algebraic plane curves

with Taketo Shirane

J. Knot Theory Ramifications 26, No. 13, Article ID 1750089, 13 p. (2017)

DOI: 10.1142/S0218216517500894

We prove that the linking set is not a homotopy invariant for algebraic plane curves, providing counterexamples to previous conjectures. This demonstrates that certain topological information about curve arrangements cannot be detected solely through homotopy-theoretic methods.

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

with Shinzo Bannai, Taketo Shirane, Hiro-O Tokunaga

Proc. Japan Acad., Ser. A 93, No. 6, 50-53 (2017)

DOI: 10.3792/pjaa.93.50

We study arrangements formed by a smooth cubic curve and its inflectional tangent lines. We analyze their topological properties and combinatorial structure, establishing connections between the geometry of the cubic and the topology of the arrangement complement.

An arithmetic Zariski pair of line arrangements with non-isomorphic fundamental group

with Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín, Miguel Marco-Buzunáriz

Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 111, No. 2, 377-402 (2017)

DOI: 10.1007/s13398-016-0298-y

We present an arithmetic Zariski pair of line arrangements whose fundamental groups are not isomorphic. This 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. This demonstrates that fundamental group information can distinguish arrangements with the same combinatorics.

On the minimal degree of logarithmic vector fields of line arrangements

with Juan Viu-Sos

Monografías Matemáticas "García de Galdeano" 40, 61-66 (2016)

We investigate the minimal degree of logarithmic vector fields associated to line arrangements and establish bounds related to the combinatorial structure. We provide dynamical interpretation and characterize polynomial vector fields having an infinite number of invariant lines.

An arithmetic Zariski 4-tuple of twelve lines

Geom. Topol. 20, No. 1, 537-553 (2016)

DOI: 10.2140/gt.2016.20.537

Using the I-invariant, 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 pairs of arrangements among this 4-tuple form new arithmetic Zariski pairs.

On complex line arrangements and their boundary manifolds

with V. Florens, M. A. Marco-Buzunariz

Math. Proc. Camb. Philos. Soc. 159, No. 2, 189-205 (2015)

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 closed regular neighbourhood 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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