Tropicalization of the Hurwitz space and universal linear-series parameter space

Determine whether, for degrees d ≤ ⌈g/2⌉, the generalized cone complex of full-rank tropical morphisms from genus-g metric graphs to metric trees is the tropicalization of the Hurwitz space of degree-d covers of the projective line, and, in the even-genus case g = 2g′ with d = g′ + 1, whether it tropicalizes the Eisenbud–Harris universal parameter space of linear series with algebraic cycle multiplicities equal to the determinantal multiplicities defined for the tropical morphisms.

Background

The paper constructs a generalized cone complex $\TM d g$ parametrizing degree-$d$ tropical morphisms from genus-$g$ metric graphs, up to tropical modification, to metric trees. For even genus $g=2g'$ and degree $d=g'+1$, the projection $\Pi\colon \TM d g\to\MTrop g$ is proved to be a surjective branched cover of degree equal to the Catalan number, with fibers counted using natural determinantal multiplicities.

The unresolved issue is whether this tropical parameter space has a direct non-Archimedean algebraic origin. Specifically, the question asks whether it is the tropicalization of the Hurwitz space of degree-d covers of $\PP^1$, whose image in the tropical moduli space would correspond to the tropicalization of the $d$-gonal locus, and whether in the even-genus case it tropicalizes the universal parameter space of linear series studied by Eisenbud and Harris. A positive answer would also identify the tropical determinantal multiplicities with the corresponding algebraic cycle multiplicities.