A tropical count of real bitangents to plane quartic curves

TL;DR

This paper provides a tropical geometric approach to count real bitangents of smooth plane quartic curves, demonstrating that certain lifting conditions are deformation-independent and offering a new proof of classical results.

Contribution

It introduces a tropical perspective on counting real bitangents, proving the independence of lifting conditions from deformations, and offers a tropical proof of classical algebraic geometry results.

Findings

Seven tropical bitangent classes for smooth tropical quartics

Lifting conditions are deformation-independent

Tropical proof of the count of real bitangents to smooth plane quartics

Abstract

A smooth tropical quartic curve has seven tropical bitangent classes. Their shapes can vary within the same combinatorial type of curve. We study deformations of these shapes and we show that the conditions determined by Cueto and Markwig for lifting them to real bitangent lines are independent of the deformations. From this we deduce a tropical proof of Pl\"{u}cker and Zeuthen's count of the number of real bitangents to smooth plane quartic curves.