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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
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