5,407 research outputs found
Lifting harmonic morphisms II: tropical curves and metrized complexes
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
Combinatorial modulus and type of graphs
Let a be the 1-skeleton of a triangulated topological annulus. We
establish bounds on the combinatorial modulus of a refinement , formed by
attaching new vertices and edges to , that depend only on the refinement and
not on the structure of itself. This immediately applies to showing that a
disk triangulation graph may be refined without changing its combinatorial
type, provided the refinement is not too wild. We also explore the type problem
in terms of disk growth, proving a parabolicity condition based on a
superlinear growth rate, which we also prove optimal. We prove our results with
no degree restrictions in both the EEL and VEL settings and examine type
problems for more general complexes and dual graphs.Comment: 24 pages, 12 figure
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