Polynomiality, wall crossings and tropical geometry of rational double hurwitz cycles

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Date
2012-12-04MFO Scientific Program
Research in Pairs 2012Series
Oberwolfach Preprints;2012,13Author
Bertram, Aaron
Cavalieri, Renzo
Markwig, Hannah
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Abstract
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and “modular” description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality between classical and tropical Hurwitz theory.