Algebraic and tropical Hurwitz cycles

The rational double Hurwitz number H_0(x) counts covers of the projective line by genus 0 curves with a given ramification profile over 0 and infinity. Hurwitz cycles H_k(x) are a higher-dimensional generalization of this: They are loci of marked rational curves admitting a cover of P^{^1} with given ramification profile over 0 and infinity and simple ramification over the marked points. Betram, Cavalieri and Markwig introduced a tropical version of these cycles by translating a certain representation of H_k(x) via intersection products to tropical vocabulary. They also show that there is a natural correspondence between faces of the tropical Hurwitz complex and strata in the classical Hurwitz cycle.

One major question that naturally comes up, is whether the tropical Hurwitz cycle can be written as an actual tropicalization of the classical one or is at least in some sense equivalent to it. We approach the problem by first studying properties of the tropical Hurwitz cycle itself. In this talk we will first briefly recap the definition of classical and tropical Hurwitz cycles. We will then discuss the combinatorial structure of the tropical cycles, i.e. what kind of objects they represent. Finally, we will present some results on tropical Hurwitzc cycles, concerning irreducibility, connectedness and how to cut them out via rational functions.