Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles

TL;DR

This paper explores the piecewise polynomial structure and wall crossings of rational double Hurwitz cycles, establishing a duality between classical and tropical geometry in the context of ramified covers of the projective line.

Contribution

It generalizes known phenomena for double Hurwitz numbers to rational double Hurwitz cycles, providing explicit descriptions of chambers and wall crossings in both classical and tropical settings.

Findings

Piecewise polynomiality of Hurwitz cycles in ramification data

Explicit description of chambers and wall crossings

Duality between classical and tropical Hurwitz theory

Abstract

We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and "modular" description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.