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 $k\in Z_{⩾1}$. 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 cyclic 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.