Tropical Hurwitz Spaces

TL;DR

This paper constructs a tropical analogue of Hurwitz spaces as polyhedral complexes, linking tropical geometry with classical enumerative geometry by encoding Hurwitz numbers through a tropical morphism.

Contribution

It introduces a tropical Hurwitz space as a connected polyhedral complex with a morphism to tropical moduli space, capturing Hurwitz numbers in a tropical setting.

Findings

Tropical Hurwitz space is a connected polyhedral complex.

The morphism degree encodes classical Hurwitz numbers.

Bridges tropical and algebraic geometry through moduli spaces.

Abstract

Hurwitz numbers are a weighted count of degree d ramified covers of curves with specified ramification profiles at marked points on the codomain curve. Isomorphism classes of these covers can be included as a dense open set in a moduli space, called a Hurwitz space. The Hurwitz space has a forgetful morphism to the moduli space of marked, stable curves, and the degree of this morphism encodes the Hurwitz numbers. Mikhalkin has constructed a moduli space of tropical marked, stable curves, and this space is a tropical variety. In this paper, I construct a tropical analogue of the Hurwitz space in the sense that it is a connected, polyhedral complex with a morphism to the tropical moduli space of curves such that the degree of the morphism encodes the Hurwitz numbers.