Divides with cusps and Kirby diagrams for line arrangements

TL;DR

This paper describes a handle decomposition and Kirby diagram for the complement of complexified real line arrangements, introducing divides with cusps as a generalization of A'Campo's divides, to better understand the topology of these affine surfaces.

Contribution

It introduces the concept of divides with cusps, extending A'Campo's divides, to facilitate the construction of Kirby diagrams for line arrangement complements.

Findings

Handle decomposition for line arrangement complements is described.

Kirby diagrams are constructed using divides with cusps.

Provides a new method to analyze the topology of affine surfaces.

Abstract

The complement of a complexified real line arrangement is an affine surface. It is classically known that such a space has a handle decomposition up to $2$-handles. We will describe the handle decomposition induced from Lefschetz hyperplane section theorem for such a space. To describe the Kirby diagram, we introduce the notion of the divide with cusps which is a generalization of the divide introduced by A'Campo.