On arrangements of smooth plane quartics and their bitangents

TL;DR

This paper explores the geometric arrangements of bitangents to smooth plane quartics, analyzing their combinatorics, automorphism groups, and constructing new examples of plane curves related to these quartics.

Contribution

It provides a detailed combinatorial analysis of bitangent arrangements, establishes bounds based on automorphism groups, and constructs new plane curves using quartics and their bitangents.

Findings

Lower bound on quadruple intersection points for certain quartic arrangements

Classification of arrangements based on automorphism group order

New examples of 3-syzygy reduced plane curves

Abstract

In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic curves. We consider quartic curves from the point of view of the order of their automorphism groups, in order to establish a lower bound on the number of quadruple intersection points for arrangements of bitangents associated with smooth plane quartics, which are smooth members of Ciani's pencil. We then construct new examples of $3$-syzygy reduced plane curves using smooth plane quartics and their bitangents.