Realizability of tropical pluri-canonical divisors

Abstract

Consider a pair consisting of an abstract tropical curve and an effective divisor from the linear system associated to $k$ times the canonical divisor for $k\in {\mathbb{Z}}_{\ge 1}$. In this article we give a purely combinatorial criterion to determine if such a pair arises as the tropicalization of a pair consisting of a smooth algebraic curve over a non-Archimedean field with algebraically closed residue field of characteristic 0 together with an effective pluri-canonical divisor. To do so, we introduce tropical normalized covers as special instances of tropical Hurwitz covers and reduce the realizability problem for pluri-canonical divisors to the realizability problem for normalized covers. Our main result generalizes the work of Möller-Ulirsch-Werner on realizability of tropical canonical divisors and incorporates the recent progress on compactifications of strata of $k$-differentials in the work of Bainbridge-Chen-Gendron-Grushevsky-Möller.

arXiv commentImproved exposition of introduction, added various clarifications and fixed typos throughout, fixed mistake in proof of Proposition 5.18, and updated bibliography

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• Functorial tropicalization of logarithmic schemes: The case of constant coefficients
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• diffstrata -- a Sage package for calculations in the tautological ring of the moduli space of Abelian differentials