Tropical Hurwitz Numbers

TL;DR

This paper develops a tropical geometric framework for Hurwitz numbers, showing that tropical analogs of branch maps accurately recover classical counts of algebraic covers.

Contribution

It introduces a tropical version of the branch map and proves its degree matches classical Hurwitz numbers, bridging algebraic and tropical geometry.

Findings

Tropical branch map degree equals classical Hurwitz number

Tropical curves serve as combinatorial models for algebraic covers

Provides a new computational approach for Hurwitz numbers

Abstract

Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers.