A generalisation of simple Harnack curves

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 (C∗)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\mathbb {C}}^{*})^{2}$$\\end{document} with totally real logarithmic Gauss map. First, we investigate which of the many properties of simple Harnack curves survive this generalisation. Then, we construct new examples using tropical geometry. Eventually, since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we determine the topological classification of such curves for any given degree.