Tropical geometry of genus two curves

TL;DR

This paper explores the tropical and analytic aspects of genus two curves, providing combinatorial rules, explicit algorithms, and invariants to understand their structure and moduli in tropical geometry.

Contribution

It introduces a combinatorial rule for dual graphs, an explicit harmonic map algorithm, and new invariants for the tropical moduli space of genus two curves.

Findings

A combinatorial rule for dual graphs and metric structures.

An explicit harmonic 2-to-1 map to a metric tree.

New invariants for the tropical moduli space.

Abstract

We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on the associated minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of P^1 with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding. Finally, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates in the tropical moduli space, we propose an alternative set of invariants that provides new length data.