Tropicalizing the space of admissible covers

TL;DR

This paper establishes a connection between tropical and classical Hurwitz moduli spaces, demonstrating that tropicalization preserves key geometric and enumerative properties, including Hurwitz numbers.

Contribution

It introduces a moduli space of tropical admissible covers and proves the compatibility of tropical and classical maps using non-archimedean geometry techniques.

Findings

Tropical and classical Hurwitz numbers are equal.

Tropicalization preserves the degree of the branch map.

A new moduli space of tropical admissible covers is constructed.

Abstract

We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We define and construct a moduli space of tropical admissible covers, and study its relationship with the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.