Tropical cycles of discrete admissible covers

TL;DR

This paper develops a tropical geometric framework for discrete admissible covers, constructing associated cycles and generalizing results on tropical curve gonality, with applications to counting covers related to Catalan numbers.

Contribution

It introduces a new tropical cycle construction for discrete admissible covers and extends gonality results to a broader tropical setting.

Findings

Constructed a polyhedral space parameterizing discrete admissible covers.

Established an equivariant cycle associated with these covers.

Derived a counting formula involving Catalan numbers for certain tropical covers.

Abstract

This article applies the technical framework developed in previous work by the author to discrete admissible covers and their moduli spaces. More precisely, we construct a poic-space that parameterizes the discrete admissible covers after fixing the genus of the target, the number of marked legs, and prescribing the ramification profiles above these marked legs. We then construct a linear poic-fibration over this poic-space and show that the usual weight assignment of covers produces an equivariant cycle of this poic-fibration. This poic-fibration comes naturally with source and target maps, and after taking the weak pushforward in top dimension through the source map and subsequently forgetting the marking, this gives rise to an equivariant tropical cycle of the corresponding spanning tree fibration. Through this framework we obtain a generalization of previously known results on these cycles pertaining to the gonality of tropical curves. Special cases of these tropical cycles yield the following result: \emph{A generic genus-$g$ $r$-marked tropical curve $\Gamma$ (with $g+r$ even) has $C_{\frac{g+r}{2}}$ many $r$-marked discrete admissible covers of degree $\frac{g+r}{2}+1$ of a tree (counted with multiplicity) that have a tropical modification of $\Gamma$ as a source, where $C_{\frac{g+r}{2}}$ is the $(\frac{g+r}{2})$th Catalan number}.