A generalisation of simple Harnack curves

TL;DR

This paper generalizes simple Harnack curves to include curves with arbitrary singularities, explores their properties using tropical geometry, and classifies those with a single hyperbolic node.

Contribution

It introduces a broader class of real algebraic curves with totally real logarithmic Gauss maps and analyzes their properties and classifications.

Findings

Generalized Harnack curves can have arbitrary singularities.

Tropical geometry helps construct examples of these curves.

Curves with a single hyperbolic node are topologically classified.

Abstract

In this paper, we suggest the following generalisation of Mikhalkin's simple Harnack curves: a generalised simple Harnack curve is a parametrised real algebraic curve in $(\mathbb{C}^*)^2$ with totally real logarithmic Gauss map. We investigate which of the many properties of simple Harnack curves survive the latter generalisation. We also show how tropical geometry allows to construct plenty of examples. Since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition where only real isolated double points can appear, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we give their topological classification as in \cite{Mikh} and show how such curves can be recovered from their spine.