2025/03/07 by Glynn, Darragh
#14H10 (Primary) #14T99 #37F34 #57K20 (Secondary) #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2503.05688
Let H be a Hurwitz space that parametrises holomorphic maps to ℙ1. Abramovich, Corti and Vistoli, building on work of Harris and Mumford, describe a compactification H with a natural boundary stratification. We show that the irreducible strata of 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 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 H, and discuss applications to complex dynamics.