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Tropical covers of curves and their moduli spaces. (English) Zbl 1318.14060
Let $$\mathcal{L}$$ be the abstract curve that corresponds to a generic tropical line in the tropical projective plane, i.e. a curve with one vertex that one denotes by $$c$$ and three ends adjacent to $$c$$ called $$u$$, $$v$$ and $$w$$. Let $$h: \Gamma \to \mathcal{L}$$ be a cover of degree $$d$$. The weights of the ends mapping to $$u$$, $$v$$ and $$w$$ give rise to partitions $$\Delta_u$$, $$\Delta_v$$ and $$\Delta_w$$ of $$d$$, and the triple $$\Delta = (\Delta_u, \Delta_v, \Delta_w)$$ is called the ramification profile of $$h$$. Now the authors introduce $$\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)$$, the moduli space of tropical covers of $$\mathcal{L}$$ of genus $$g$$ with ramification profile $$\Delta$$.
The tropical branch map on $$\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)$$ is $$\mathrm{br}^{\mathrm{trop}}: \mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta) \to \mathcal{L}^r$$, $$(h: \Gamma \to \mathcal{L}) \mapsto (h(V_1),h(V_2),\dots,h(V_r))$$, with $$r := \#\Delta + 2g - 2 - d$$ the total number of labels.
The main theorems are:
Theorem 2.15. The moduli space $$\mathrm{M}_g^{\mathrm{trop}}(\mathcal{L},\Delta)$$ is an abstract weighted polyhedral complex of pure dimension $$r$$.
Theorem 3.3. The degree of $$\mathrm{br}^{\mathrm{trop}}$$ is constant, called the tropical Hurwitz number $$H_d^{g,\mathrm{trop}}(\Delta)$$.
Theorem 3.6. The tropical Hurwitz numbers $$H_d^{g,\mathrm{trop}}(\Delta)$$ defined using tropical intersection theory equal their algebraic counterparts $$H_d^g(\Delta)$$.

##### MSC:
 14T05 Tropical geometry (MSC2010) 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 51M20 Polyhedra and polytopes; regular figures, division of spaces
##### Keywords:
tropical geometry; Hurwitz numbers; covers of curves
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