Galois quotients of tropical curves and invariant linear systems
type: Item
instance of: scholarly article
title: Galois quotients of tropical curves and invariant linear systems
author: Juae Song
published in: Hokkaido Mathematical Journal
publication date: 2022-12-08T00:00:00Z
review text: In algebraic geometry, embeddings of curves into projective space are defined via linear systems. A similar concept is known in tropical geometry, where the notion of an algebraic curve gets replaced by the notion of a \textit{tropical curve}, i.e. a finite metric graph. More concretely, given a tropical curve \(\Gamma\), a divisor \(D\), and a finite generating set \(F = \{f_0, \ldots, f_n\}\) of the space \(R(D)\) consisting of piecewise linear functions with pole orders bounded by \(D\), then we obtain a rational map \begin{align*} \phi_F : \Gamma &\longrightarrow \mathbb{T}\mathbb{P}^n \\ x &\longmapsto \big(f_0(x) : \ldots : f_n(x) \big). \end{align*} In the present article, the authors studies this situation under the additional information of an action of a finite group \(K\) on \(\Gamma\). Similar to the above, a finite generating set \(F = \{f_i\}_i\) of the space of \(K\)-invariant PL functions \(R(D)^K \subseteq R(D)\) gives a rational map \(\phi_F\) to tropical projective space. As a first theorem, the author shows that \(\phi_F\) exhibits \(\Gamma\) as a \(K\)-Galois cover of \(\phi_F(\Gamma)\) if and only if \(\phi_F\) is injective on the level of \(K\)-orbits. Here, a \textit{\(K\)-Galois cover} of tropical curves is a harmonic morphism \(\Gamma \to \Gamma'\) of degree \(|K|\), together with a \(K\)-action on \(\Gamma\), such that the morphism is the quotient map modulo \(K\). As a second theorem, the author shows that the required injectivity is actually automatic as long as \(D\) is effective, \(K\)-invariant, and of positive degree. As an application, the authors studies tropical hyperelliptic curves, i.e. curves which posses a \(g^1_2\). In the classical setting, the canonical map \(\phi_{|K_C|}\) of an algebraic curve \(C\) which is given by the linear system of the canonical divisor \(K_C\) gives rise to a double cover \(C \to \mathbb{P}^1\). While tropically, a \(g^1_2\) gives rise to a harmonic double cover of a metric tree (the tropical analo…
reviewed by: Felix Röhrle
Mathematics Subject Classification ID: 14T15
Mathematics Subject Classification ID: 14T20
zbMATH DE Number: 7628898
zbMATH Keywords: tropical curve
zbMATH Keywords: invariant linear subsystem
zbMATH Keywords: rational map
zbMATH Keywords: Galois covering
zbMATH Keywords: hyperelliptic tropical curve
zbMATH Keywords: canonical map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/2020-406
OpenAlex ID: W4312767715
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cites work: Q3533399
DOI: 10.14492/HOKMJ/2020-406
description: scientific article; zbMATH DE number 7628898