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

TL;DR

This paper proves a multiplicativity property of the $ ext{I}$-invariant for arrangements when glued along a triangle, leading to new Zariski pair constructions with different topologies but identical combinatorics.

Contribution

It establishes a multiplicativity theorem for the $ ext{I}$-invariant under arrangement gluing and applies it to produce new examples of Zariski pairs.

Findings

Proves multiplicativity of the $ ext{I}$-invariant under gluing arrangements.

Shows extended Rybnikov arrangements form ordered Zariski pairs.

Provides a method to construct new Zariski pairs using the $ ext{I}$-invariant.

Abstract

The invariant $\mathcal{I}(\mathcal{A},\xi,\gamma)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnikov arrangements form an ordered Zariski pairs (i.e. two arrangements with the same combinatorial information and different ordered topologies). Finally, we extend this method to a particular family of arrangements and thus we obtain a method to construct new examples of Zariski pairs.