Tropical Igusa Invariants

TL;DR

This paper provides a new proof for determining the minimal Berkovich skeleton of genus two curves over non-archimedean fields using Igusa invariants, and interprets these results through tropical geometry.

Contribution

It offers a novel proof applicable to all complete non-archimedean fields and connects Igusa invariants with tropical moduli spaces.

Findings

Criteria for Igusa invariants determining skeletons

Tropicalization map factors through a concrete embedding

Interpretation of Igusa invariants in tropical geometry

Abstract

Let $X$ be a smooth geometrically connected projective curve of genus two over a complete non-archimedean field $K$. For discretely valued $K$, the first main theorem in \cite{liu} gives a set of criteria on the Igusa invariants of the curve that determine the minimal Berkovich skeleton of $X$ together with its edge lengths and vertex weights. In this paper we use the theory of Berkovich spaces to give a new proof of this theorem that works for arbitrary complete non-archimedean fields. We furthermore interpret the final result in terms of tropical moduli spaces and tropical Igusa invariants. This reformulation shows that the abstract tropicalization map ${M}_{2}\to\mathrm{trop}(M_{2})$ factors through the tropicalization of a concrete embedding of ${M}_{2}$ into a weighted projective space.