Tropical Tevelev degrees

Abstract

Tropical Hurwitz spaces parameterize genus g, degree d covers of a tropical rational curve with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product Mgnbar{g,n} cross Mgnbar{0,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. We define the tropical Tevelev degrees, Tev_g^trop in analogy to the algebraic case. We develop an explicit combinatorial construction that computes Tev_g^trop = 2^g. We prove that these tropical enumerative invariants agree with their algebraic counterparts, giving an independent tropical computation of the algebraic degrees Tev_g. We finally generalize tropical Tevelev degrees to more cases and construct computations of these invariants.