Abstract
Hurwitz numbers count genus g, degree d covers of ℙ1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43–92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79–96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0.
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P. Johnson was supported in part by University of Michigan RTG grant 0602191.
H. Markwig was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
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Cavalieri, R., Johnson, P. & Markwig, H. Tropical Hurwitz numbers. J Algebr Comb 32, 241–265 (2010). https://doi.org/10.1007/s10801-009-0213-0
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DOI: https://doi.org/10.1007/s10801-009-0213-0