Tropical Intersection Theory on Moduli Stack of Curve Coverings

Abstract

We construct the moduli cone stack $\mathfrac{M}\eta^\text{trop}$ of tropical '{e}tale covers (i.e., coverings of twisted tropical curves). We define the tropical intersection theory on $\mathfrac{M}\eta^\text{trop}$ and show that the tropical intersection theory agrees with the intersection theory on the moduli stack $\bar{\mathfrac{M}}\eta$ of '{e}tale covers (i.e., coverings of twisted algebraic curves). We apply the tropical intersection theory on $\mathfrac{M}\eta^\text{trop}$ to calculate the intersection numbers of Psi-classes on the moduli space $\bar{\mathfrac{M}}{g,n}$ of $n$-marked genus $g$ curves. We also define the moduli stack $\mathfrac{M}\eta^\text{log}$ of logarithmic '{e}tale covers and describe the tropicalization map from $\mathfrac{M}\eta^\text{log}$ to the Artin fan of $\mathfrac{M}\eta^\text{trop}$.