Tropical double ramification loci

TL;DR

This paper studies the tropical geometry of the double ramification locus in moduli space, showing two definitions form generalized cone complexes and solving a key part of the realizability problem.

Contribution

It introduces two tropical analogues of the double ramification locus, proves their structure as cone complexes, and links the realizability problem to the Hurwitz existence problem.

Findings

Both loci admit a structure of generalized cone complexes.

The principal divisor locus has codimension zero in tropical moduli space.

The ramified cover locus has codimension g, matching expectations.

Abstract

Motivated by the realizability problem for principal tropical divisors with a fixed ramification profile, we explore the tropical geometry of the double ramification locus in $\mathcal{M}_{g,n}$.There are two ways to define a tropical analogue of the double ramification locus: one as a locus of principal divisors, the other as a locus of finite effective ramified covers of a tree. We show that both loci admit a structure of a generalized cone complex in $M_{g,n}^{trop}$, with the latter contained in the former. We prove that the locus of principal divisors has cones of codimension zero in $M_{g,n}^{trop}$, while the locus of ramified covers has the expected codimension $g$. This solves the deformation-theoretic part of the realizability problem for principal divisors, reducing it to the so-called Hurwitz existence problem for covers of a fixed ramification type.