Tropicalizing tame degree three coverings of the projective line

Abstract

In this paper, we study the problem of tropicalizing tame degree three coverings of the projective line. Given any degree three covering $C⟶{\mathbb{P}}^1$, we give an algorithm that produces the Berkovich skeleton of $C$. In particular, this gives an algorithm for finding the Berkovich skeleton of a genus $3$ curve. The algorithm uses a continuity statement for inertia groups of semistable Galois coverings, which we prove first. After that we give a formula for the decomposition group of an irreducible component $\Gamma \subset {\mathcal{C}}_s$ for a semistable Galois covering $\mathcal{C}⟶\mathcal{D}$. We conclude the paper with a simple application of these $S_3$-coverings to elliptic curves, giving another proof of the familiar semistability criterion for elliptic curves using a natural degree three morphism to ${\mathbb{P}}^1$ instead of the usual degree two morphism.

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