Boundary stratifications of Hurwitz spaces

Abstract: Let $\mathcal{H}$ be a Hurwitz space that parametrises holomorphic maps to $\mathbb{P}^1$. Abramovich, Corti and Vistoli, building on work of Harris and Mumford, describe a compactification $\overline{\mathcal{H}}$ with a natural boundary stratification. We show that the irreducible strata of $\overline{\mathcal{H}}$ are in bijection with combinatorial objects called decorated trees (up to a suitable equivalence), and that containment of irreducible strata is given by edge contraction of decorated trees. This combinatorial description allows us to define a tropical Hurwitz space, refining a definition given by Cavalieri, Markwig and Ranganathan. We also discuss applications to complex dynamics.