Configurations of points and topology of real line arrangements

Abstract

A central question in the study of line arrangements in the complex projective plane \(\mathbb {C}\mathbb {P}^2\) 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 \(\mathbb {R}\mathbb {P}^2\). Using this dual setting, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in \(\mathbb {C}\mathbb {P}^2\) (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 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.