Combinatorics of tropical Hurwitz cycles

TL;DR

This paper investigates properties of tropical Hurwitz cycles, demonstrating their connectivity, irreducibility under certain conditions, and their representation as divisors of rational functions, advancing tropical intersection theory.

Contribution

It establishes the connectedness and weak irreducibility of tropical Hurwitz loci and links these cycles to tropical boundary divisors, providing new insights into their structure.

Findings

All tropical Hurwitz loci are connected in codimension one.

Under generic conditions, Hurwitz cycles are weakly irreducible.

Hurwitz cycles are numerically equivalent to tropical boundary divisors.

Abstract

We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtained with the help of a-tint, an extension for polymake for tropical intersection theory.