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 $k \in \mathbb{Z}_{\geq 1}$. 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.

Figures (13)

• Figure 1: Overview of the notions and notation used in this article. Horizontal arrows forget the cover. The commutative triangle in \ref{['eq:reduction_triangle']} can be seen on the outside of this diagram.
• Figure 2: Graph $G$ with divisor $D$ defining a cone in $\mathbb{P} \Omega^3 M_g^{\operatorname{trop}}$ of maximal dimension.
• Figure 3: Graph $G$ with divisor $D'$ defining a cone in $\mathbb{P} \Omega^3 M_g^{\operatorname{trop}}$ which is maximal but not of maximal dimension.
• Figure 4: Reduced types $\mu_\mathrm{red}$ and generators $w_i$ of the $\mathbb{C}$-lines missing in the image of the residue map.
• Figure 5: A cover of graphs with the action of the deck-transformation $\tau$.

Theorems & Definitions (99)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• proof : Proof of Theorem \ref{['intro:thm:trop_k_Hodge_bundle']}