In this paper we prove several lifting theorems for morphisms of tropical
curves. We interpret the obstruction to lifting a finite harmonic morphism of
augmented metric graphs to a morphism of algebraic curves as the non-vanishing
of certain Hurwitz numbers, and we give various conditions under which this
obstruction does vanish. In particular we show that any finite harmonic
morphism of (non-augmented) metric graphs lifts. We also give various
applications of these results. For example, we show that linear equivalence of
divisors on a tropical curve C coincides with the equivalence relation
generated by declaring that the fibers of every finite harmonic morphism from C
to the tropical projective line are equivalent. We study liftability of
metrized complexes equipped with a finite group action, and use this to
classify all augmented metric graphs arising as the tropicalization of a
hyperelliptic curve. We prove that there exists a d-gonal tropical curve that
does not lift to a d-gonal algebraic curve.
This article is the second in a series of two.Comment: 35 pages, 18 figures. This article used to be the second half of
arXiv:1303.4812, and is now its seque
