Tropical double ramification loci

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.

arXiv comment68 pages

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• The moduli space of stable quotients
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Cited by in corpus (1)

• A case study of intersections on blowups of the moduli of curves