Tropical covers of curves and their moduli spaces

TL;DR

This paper develops a tropical geometric framework for studying covers of curves, defining moduli spaces and branch maps, and proves invariance of their degree to justify tropical counts of Hurwitz numbers.

Contribution

It introduces a tropical moduli space of covers, proves the invariance of the branch map degree, and links tropical intersection theory with classical Hurwitz number computations.

Findings

Degree of the tropical branch map is invariant.

Tropical intersection-theoretic definition of Hurwitz numbers matches classical results.

Provides a duality between tropical moduli spaces and relative stable maps.

Abstract

We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersection-theoretic definition of tropical triple Hurwitz numbers. We show that our intersection-theoretic definition coincides with the one given by Bertrand, Brugall\'e and Mikhalkin in the article "Tropical Open Hurwitz numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus we provide a tropical intersection-theoretic justification for the multiplicities with which a tropical cover has to be counted. Our method of proof is to establish a local duality between our tropical moduli spaces and certain moduli spaces of relative stable maps to the projective line.