Galois quotients of tropical curves and invariant linear systems
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 analogue of \(\mathbb{P}^1\)), the canonical map will not always be of this form. In fact, the author shows that the canonical map is a double cover of a tree precisely when \(g(\Gamma) = 2\). Furthermore, the authors shows that if \(\Gamma\) is hyperelliptic and \(g(\Gamma) \geq 3\), then a double cover of the desired form is given by the part of the canonical linear system which is invariant under the hyperelliptic involution. A noteworthy technical detail is that since \(\phi_F : \Gamma \to \phi_F(\Gamma)\) is always an isometry, the author has to introduce multiplicities on the edges of \(\Gamma\), given by the size of their stabilizer groups, in order to make the map harmonic.
scientific article; zbMATH DE number 7628898
scientific article