Fundamental Groups of Real Arrangements and Torsion in the Lower Central Series Quotients

Bartolo, Enrique Artal; Guerville-Ballé, Benoît; Viu-Sos, Juan

doi:10.1080/10586458.2018.1428131

Cited by 8 publications

(11 citation statements)

References 27 publications

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“…In an upcoming paper [ABGBVS17], the authors prove that some of the Zariski pairs presented in this paper can be distinguished by their fundamental groups. More precisely, the corresponding lower central factors differ by a torsion element.…”

Section: Introductionmentioning

confidence: 86%

Configurations of points and topology of real line arrangements

Guerville-Ballé

Viu-Sos

2018

Math. Ann.

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A central question in the study of line arrangements in the complex projective plane CP 2 is the following: 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. This invariant is based on the weight counting over the points of the associated dual configuration, located in particular chambers of the real projective plane RP 2 .Using this dual setting, 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 our examples, we derive some degenerations containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.We compute explicitly 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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Section: Introductionmentioning

confidence: 86%

Configurations of points and topology of real line arrangements

Guerville-Ballé

Viu-Sos

2018

Math. Ann.

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“…Due to their particular arithmetic property, it would be interesting to see if the invariants developed by Bannai, Shirane and Tokunaga [6,22,25] could distinguish their topology. Furthermore, neither the linking-invariants [7,13] nor the torsion order of the first lower central series quotients of their fundamental groups [23,8] can distinguish it.…”

Section: Supportmentioning

confidence: 99%

“…As mentioned in the abstract, the questions related to the combinatorial nature of some properties of a hyperplane arrangement are numerous in the literature. If some of them have been solved by the affirmative, as for the number of chambers of a real arrangement [27], the cohomology ring of the complement [19], the rank of the lower central series quotients of the fundamental group of its complement [10] or the deletion and addition-deletion theorems of free arrangements [2,1]; some others obtained a negative answer, as for the embedded topology of a complex arrangements or the fundamental group of its complements, see [21,12,5,13], (also negative for the smaller class of real complexified arrangements [3,14]), the torsion of the lower central series quotients [8] or the existence of unexpected curves [14]. Naturally, the number of problems which are still open (or conjectural) is larger; like the famous Terao's conjecture [24,20], the combinatorial nature of the characteristic varieties [15] or of the homology of the Milnor fiber [11,Problem 4.5], to name some but a few.…”

Section: Introductionmentioning

confidence: 99%

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

Guerville-Ballé¹

2021

Preprint

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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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“…Nevertheless, as a consequence of Theorem 4.3 and the preceding discussion, we have the following result. Consequently, the group gr 3 (G(A)) is combinatorially determined; that is, if A and B are two arrangements such that L ≤2 (A) L ≤2 (B), then gr 3 (G(A)) gr 3 (G(B)).…”

Section: Hyperplane Arrangementsmentioning

confidence: 99%

Homology, lower central series, and hyperplane arrangements

Porter

Suciu

2019

European Journal of Mathematics

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We explore finitely generated groups by studying the nilpotent towers and the various Lie algebras attached to such groups. Our main goal is to relate an isomorphism extension problem in the Postnikov tower to the existence of certain commuting diagrams. This recasts a result of G. Rybnikov in a more general framework and leads to an application to hyperplane arrangements, whereby we show that all the nilpotent quotients of a decomposable arrangement group are combinatorially determined.

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Fundamental Groups of Real Arrangements and Torsion in the Lower Central Series Quotients

Cited by 8 publications

References 27 publications

Configurations of points and topology of real line arrangements

Configurations of points and topology of real line arrangements

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

Homology, lower central series, and hyperplane arrangements

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