Tropical Hurwitz Numbers

Abstract

Hurwitz numbers count genus g, degree d covers of the projective line 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.

Published in Journal of Algebraic Combinatorics, Volume 32, Number 2 / September, 2010. Added section on genus zero piecewise polynomiality. Removed paragraph on psi classes