A topological invariant of line arrangements

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.

Section: Supportmentioning

confidence: 99%

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

2021

Preprint

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“…According to [7,Lemma 4.3] which describes the map j : H 1 (B( Ã)) → H 1 (M ( Ã)) (see also [17, Theorem 4.3]) for cycles supported by three lines (here F 0 , L and F P ), we obtain…”

Section: Topological Computationmentioning

confidence: 99%

“…Using the notation 10 described in Section 3.1, we have the following braided wiring diagrams 11 for M 1 and M 3 , they are also pictured in Figure 1 and 2. [8,6,7,5]], [(), [8,2]], [(), [8,1,10]], [(), [8,9]], [(), [8,4,3]],…”

Section: Ordered Zariski Pair With Ten Linesmentioning

confidence: 99%

“…We should also mention the discovery of 11 new Zariski pairs [19,22] or the first family of arithmetic Zariski tuple [12]. All these recent results arise from two papers: the construction of a linking invariant for line arrangements [7] (called the I-invariant), and the description of the inclusion of the boundary manifold 4 in the complement of A [17] (which allows to compute the I-invariant). Therefore, the study of the linking properties of line arrangements and the resulting Zariski pairs could help solve open problems like the combinatoriality of the characteristic varieties or the questions related to the Milnor fiber.…”

Section: Introductionmentioning

confidence: 99%

“…associated to the tensor Λ, defined as L (A, Λ) = π • Ψ(Λ), is an invariant of the ordered and oriented topology 6 of A, see [12,Proposition 21]. It generalizes the I-invariant, in the sense that this former linking invariant is given by I(A, ξ, γ) = L (A, ξ ⊗ γ), when (A, ξ, γ) is an inner-cyclic triple (see [7] for the definition).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The loop linking number of line arrangements

2020

Preprint

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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 homotopyequivalent lattice-isomorphic arrangements which have non-homeomorphic complements.

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

2018

J. Math. Soc. Japan

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The invariant I(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 pair (i.e. two arrangements with the same combinatorial information and different ordered topologies).Finally, we extend this method to a family of arrangements and thus we obtain a method to construct new examples of Zariski pairs.