2021/09/08 by Felix Röhrle, Johannes Schwab, Röhrle, Felix +1
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2109.03579
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 ∈ 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 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.