Configurations of points and topology of real line arrangements

Abstract

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.

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

“…This particular pair possesses 13 lines, 11 points of multiplicity 3, 2 of multiplicity 5 and the following combinatorics: L 1 , L 4 , L 6 , L 9 , L 13 , L 1 , L 5 , L 7 , L 1 , L 8 , L 10 , L 1 , L 11 , L 12 , L 2 , L 4 , L 7 , L 10 , L 12 , L 2 , L 5 , L 6 , L 2 , L 8 , L 9 , L 2 , L 11 , L 13 , L 3 , L 4 , L 5 , L 3 , L 6 , L 8 , L 3 , L 7 , L 11 , L 3 , L 9 , L 10 , L 3 , L 12 , L 13 , where each subset describes an incidence relation between lines; pairs which do not appear correspond to double points. It is proven in [GV17,Prop. 4.5] that Σ decomposes as topological space on two connected components Σ = Σ 0 ⊔ Σ 1 , each of them being isomorphic to a punctured complex affine line, where any pair of arrangements A ∈ Σ 0 and A ′ ∈ Σ 1 forms a Zariski pair.…”

“…In the present paper, we give a negative answer to above problems exhibiting an explicit example. Indeed, considering one of the real complexified Zariski pairs recently constructed by the last two named authors in [GV17], we prove in Corollary 2.7 that the fundamental groups of their complements are not isomorphic. More precisely, in Theorem 2.3, we prove that the 4 th and the 5 th quotient groups of the lower central series of these fundamental groups differ by a 2-torsion element (the computations, made using GAP [GAP17], mainly the package nq [HN16], are described in Appendix 3).…”

“…In [GV17], several examples of real complexified Zariski pairs are given. In particular, the modulli space of realizations Σ of one of them is studied in [GV17,Sec. 4].…”

“…It is well-known, since Rybnikov [Ryb11], that the combinatorics of A (or equivalently the underlying matroid or intersection lattice of A) does not determine its topology. Before [GV17] there were three known examples of Zariski pairs of line arrangements, i.e. a pair of intersection latticeequivalent arrangements with different topologies.…”

A linking invariant for algebraic curves

“…In the particular case of line arrangements, the linking invariant (in the form of the I-invariant) has been successfully used in [9] to detect a Zariski pair of 12 lines. Recently, the first author and Viu-Sos gave an effective diagrammatic reformulation of this invariant in the particular case of real line arrangements, see [11]. Using this reformulation, they provide 10 examples of complexified real Zariski pairs.…”