Lifting tropical bitangents

TL;DR

This paper investigates how tropical bitangents to algebraic curves lift to complex algebraic bitangents, providing constructive methods and contributing to understanding the classical 28 bitangent lines of a smooth quartic.

Contribution

It offers a constructive approach to lifting tropical bitangents and shows that generically, these lift in sets of four, advancing the tropical proof of classical algebraic geometry results.

Findings

All seven tropical bitangents of a smooth tropical plane quartic lift in sets of four.

Provides explicit solutions for the initial coefficients of the lifted bitangent lines.

Lays groundwork for counting real bitangents using tropical methods.

Abstract

We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical plane quartic lift in sets of four to algebraic bitangents. We do this constructively, i.e. we give solutions for the initial terms of the coefficients of the bitangent lines. This is a step towards a tropical proof that a general smooth quartic admits 28 bitangent lines. The methods are also appropriate to count real bitangents, however the conditions to determine whether a tropical bitangent has real lifts are not purely combinatorial.