Tropical Igusa Invariants

Let X be a smooth geometrically connected projective curve of genus two over a complete nonarchimedean field K. For discretely valued K, the first main theorem in [Liu93] gives a set of criteria on the Igusa invariants of the curve that determine the minimal 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 M2 → trop(M2) factors through the tropicalization of a concrete embedding of M2 into a weighted projective space.