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, which we identify with the skeleton of the Berkovich analytification of $\overline{\mathcal{H}}$. The tropical Hurwitz space that we obtain is a refinement of a version defined by Cavalieri, Markwig and Ranganathan. We also provide an implementation that computes the stratification of $\overline{\mathcal{H}}$, and discuss applications to complex dynamics.