×

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.

MSC:

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

References:

[1] Baker, M., Faber, X.: Metrized graphs, Laplacian operators, and electrical networks. In: Quantum graphs and their applications, Contemp. Math., vol. 415, pp. 15–33. Am. Math. Soc., Providence, RI (2006) (see also Metrized graphs, electical networks, and Fourier analysis. Available at: http://arxiv.org/abs/math/0407428 ) · Zbl 1114.94025
[2] Baker, M., Rumely, R.: Harmonic analysis on metrized graphs. Can. J. Math. 59(2), 225–275 (2007). Available at: http://arxiv.org/abs/math/0407427 · Zbl 1123.43006
[3] Goldstein, D., Guralnick, R., Joyner, D.: A question about Pic(X) as a G-module. In: Shaska, T. (ed.) Computational Aspects of Algebraic Curves, Lecture Notes in Computing. WorldScientific, USA (2005). Available at: http://front.math.ucdavis.edu/math.AG/0407036 · Zbl 1121.14020
[4] Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008). Available at: http://aps.arxiv.org/abs/math/0612129 · Zbl 1187.14066
[5] Haase, C., Musiker, G., Yu, J.: Linear systems on tropical curves (2009). Available at: http://arxiv.org/pdf/0909.3685 · Zbl 1408.14201
[6] Joyner, D.: A primer on computational group homology and cohomology. In: Fine, B. (ed.) Aspects of Infinite Groups. The Proceedings of the Gaglione conference. World Scientific Press, USA (2009) (a longer version is available at: http://front.math.ucdavis.edu/0706.0549 )
[7] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, draft of book (2009). Available at: http://www.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf
[8] Mikhalkin, G.: Tropical geometry and its applications. In: International Congress of Mathematicians, vol. II, pp. 827–852, Eur. Math. Soc., Zurich (2006). Available at: http://arxiv.org/abs/math/0601041 · Zbl 1103.14034
[9] Mikhalkin G.: What is a tropical curve?. Notices Am. Math. Soc. 54(4), 511–513 (2007) · Zbl 1142.14300
[10] Mikhalkin, G.: Enumerative tropical algebraic geometry in $${\(\backslash\)mathbb{R}\^2}$$ . J. Am. Math. Soc. 18(2), 313–377 (2005). Available at: http://arxiv.org/abs/math/0312530
[11] Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. In: Curves and abelian varieties, Contemp. Math. vol. 465, pp. 203–230, Am. Math. Soc., Providence, RI (2008). Available at: http://arxiv.org/pdf/math.AG/0612267v2 · Zbl 1152.14028
[12] Richter-Gebert, J., Sturmfels, B., Theobald, T.: First steps in tropical geometry, In: Idempotent mathematics and mathematical physics, Contemp. Math., vol. 377, pp. 289–317, Amer. Math. Soc., Providence, RI (2005). Available at: http://arxiv.org/abs/math/0306366 · Zbl 1093.14080
[13] Rotman, J.: Advanced modern algebra. Prentice Hall, Englewood Cliffs (2002) · Zbl 0997.00001
[14] Serre J.-P.: Local fields. Springer, Berlin (1979)
[15] Shatz S.: Profinite groups, arithmetic, and geometry. Princeton Univ. Press, New Jersey (1972) · Zbl 0236.12002
[16] Speyer, D., Sturmfels, B.: Tropical mathematics. Math. Mag. 3, 163–173 (2009). Available at: http://arxiv.org/abs/math/0408099 · Zbl 1227.14051
[17] Zhang S.: Admissible pairing on a curve. Invent. math. 112, 171–193 (1993) · Zbl 0795.14015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.