Tropical hyperelliptic curves

TL;DR

This paper characterizes tropical hyperelliptic curves using harmonic morphisms, describes their moduli space structure, and connects classical hyperelliptic curves with tropical geometry through Berkovich skeleta.

Contribution

It generalizes harmonic morphism characterization of hyperelliptic graphs to the metric case and describes the moduli space as a polyhedral fan.

Findings

The locus of tropical hyperelliptic curves forms a (2g-1)-dimensional stacky polyhedral fan.

Maximal cells correspond to trees with g-1 vertices of valence at most 3.

Classical hyperelliptic curves' Berkovich skeleta lie in specific maximal cells called standard ladders.

Abstract

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g-1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g-1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition lies in a maximal cell of genus g called a standard ladder.