Configurations of points and topology of real line arrangements

TL;DR

This paper introduces a new topological invariant called chamber weight for real line arrangements, which helps distinguish arrangements with identical combinatorics but different topologies, leading to new examples of Zariski pairs.

Contribution

The work defines the chamber weight invariant and demonstrates its effectiveness in distinguishing complexified real line arrangements with identical combinatorics.

Findings

Constructed new Zariski pairs of 13, 15, and 17 lines over Q.

Computed the moduli space of a specific arrangement, showing it has two connected components.

Provided geometric characterizations of the moduli space components.

Abstract

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ 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, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties. Using this dual point of view, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in $\mathbb{CP}^2$ (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 $\mathbb{Q}$ and containing only double and triple points. For each one of them, we can derive degenerations, containing points of multiplicity 2, 3 and 5, which are also Zariski pairs. We explicitly compute 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.