A note on the topology of arrangements for a smooth plane quartic and its bitangent lines

TL;DR

This paper constructs a Zariski triple of arrangements involving a smooth plane quartic and its four bitangent lines, using a matrix related to height pairings to distinguish their topologies.

Contribution

It introduces a new topological invariant based on height pairings to differentiate arrangements of quartic curves and their bitangents.

Findings

Constructed a Zariski triple of arrangements

Identified a matrix criterion for topological distinction

Demonstrated the effectiveness of height pairing in topology analysis

Abstract

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points arising from three bitangent lines.