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Tropicalizing the space of admissible covers. (English) Zbl 1373.14064
The authors provide an intersection theoretic framework for the tropical computation of Hurwitz numbers. The tropical correspondence theorems for Hurwitz numbers (claiming that the classical Hurwitz numbers equal the tropical ones in an appropriate sense) were established in [B. Bertrand, et al., Rend. Semin. Mat. Univ. Padova 125, 157–171 (2011; Zbl 1226.14066)] and [ P. Cavalieri et al., J. Algebr. Comb. 32, No. 2, 241–265 (2010; Zbl 1218.14058)]. The authors describe the tropicalization of the moduli space of admissible covers, and then deduce the tropical correspondence for Hurwitz numbers from the interpretation of the latter as the degree of a certain tautological map (the branch map) of the moduli space. The relation between the tropical and the classical moduli spaces is established via the skeleta of the Berkovich analytification of the latter, based on the technique in [D. Abramovich et al., Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765–809 (2015; Zbl 1410.14049)].

MSC:
14T05 Tropical geometry (MSC2010)
14H10 Families, moduli of curves (algebraic)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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