An arithmetic Zariski 4–tuple of twelve lines

Abstract

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 conjugate in a number field with the same combinatorial information but with different embedding topology in CP 2 .

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

“…arrangements where the lines are defined by real equation). The third known example is obtained by the author in [6]. The topologies of this example were distinguished using the invariant I(A, ξ, γ) (also called the I-invariant).…”

The height of a permutation and applications to distance between real line arrangements

“…Rybnikov [9] found the first such pair of arrangements in 1998. Bartolo et al [5] and then Guerville-Ballé [6] give other examples, with the latter example revisited by Bartolo et al [4].…”

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

“…a pair of intersection latticeequivalent arrangements with different topologies. The first one, by Rybnikov [Ryb11,ACCM07], is a pair of line arrangements admitting no real equation and distinguished by the fundamental group of the complement G A ; the last one (with coefficients conjugated in a complex Galois-extension) was first distinguished with a linking property [Gue16] and later by their fundamental group [ACGM17]; the remaining one [ACCM05] is the only example which can be realized with all lines defined by real coefficients, and its topology is distinguished by the braid monodromy, see [Chi33,Che73,Moi81] in the context of algebraic plane curves and surfaces, and [Sal88a,CS97] for more details in the case of arrangements. It is worth noticing that, in this last example, we currently do not know if their fundamental groups are isomorphic or not, but it turns out that their profinite completions are (since their equations are conjugated in a real Galoisextension).…”