Automorphism groups on tropical curves: some cohomology calculations. (English) Zbl 1286.14077

Tropical geometry is a fast growing field of research, in close interaction with classical algebraic geometry. What does tropical mean? We simply change the classical operations addition and multiplication to new ones. Given two real numbers \(a\) and \(b\), the tropical sum is the maximum \(\max\{a,b\}\) and the tropical product is \(a+b\).
Abstract tropical curves were introduced by Mikhalkin in 2005. The study of abstract tropical curves is of highly current interest, as anyone can easily check.
Let \(X\) be an abstract tropical curve and consider a finite group of automorphisms \(G\subset \mathrm{Aut}(X)\). In this paper the authors show (in theorem 3) that if \(D\) is a divisor on \(X\) such that its divisor class (in the Picard group \(Pic(X)\)) is \(G\)-invariant, then there is a \(G\)-invariant divisor \(D'\) linearly equivalent to \(D\).
In order to achieve this result, they prove (in theorem 2) that the cohomology group \(H^1(G,M(X))\) is trivial, where \(M(X)\) denotes the group of rational functions on \(X\) (here, tropical multiplication is the group operation). This is a tropical analogue of Hilbert’s 90th theorem, stated in homological algebra terms.
To do so, the authors find a short exact sequence of \(\mathbb{Z}[G]\)-modules, namely \(0\rightarrow \mathbb{R}\rightarrow M(X)\rightarrow \mathrm{Prin}(X)\rightarrow 0\), and use the usual short exact sequence \(0\rightarrow \mathrm{Prin}(X)\rightarrow \mathrm{Div}(X)\rightarrow \mathrm{Pic}(X) \rightarrow 0\). Then they compute with the corresponding long exact sequences in cohomology.
Reviewer’s remark: The definition of abstract tropical curve used in this paper is based on star-shaped sets, metric topological graphs (where every leaf has infinite length) and models of these. This definition is slightly different from other definitions in use.
There is a minor slip in figure 1: according to the definition in the paper, the star-shaped set having 3 arms is the symmetric image (with respect to a vertical axis) of what is shown in the figure.


14T05 Tropical geometry (MSC2010)
14H37 Automorphisms of curves
Full Text: DOI arXiv


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