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Tropical hyperelliptic curves. (English) Zbl 1266.14050

Tropical geometry is a fairly recent trend in mathematics. It can be viewed as a degeneration of algebraic geometry. Therefore, there are fascinating connections between algebraic and tropical varieties, some of them yet to be discovered.
This paper brings together the tropical Brill–Noether theory (due to Baker, Caporaso, Cools, Draisma, Payne, Robeva, Lim, Potashnik, etc., since 2008) and moduli spaces of tropical curves (due to Mikhalkin, Zharkov, Caporaso, since 2006).
In this paper the author studies the locus of tropical hyperelliptic curves as a subset of the moduli space of tropical curves, for a fixed genus \(g\). A more general problem is to characterize the tropical Brill–Noether loci (of tropical curves of genus \(g\) admitting a divisor of degree \(d\) and rank at least \(r\), when \(g-(r+1)(g-d+r)<0\)), the hyperelliptic case being \(d=2\), \(r=1\).
M. Baker and S. Norine studied combinatorial graphs since 2007 [Adv. Math. 215, No. 2, 766–788 (2007; Zbl 1124.05049); Int. Math. Res. Not. 2009, No. 15, 2914–2955 (2009; Zbl 1178.05031)]. This notion evolved into that of metric graph, in close relation with the concept of abstract tropical curve. The author gives a new definition of harmonic morphism between metric graphs and proves that a metric graph \(\Gamma\) is a tropical hyperelliptic curve if and only if there exists a degree 2 harmonic morphism from \(\Gamma\) to a metric tree.
It is well–known that, for genus \(g\), the classical hyperelliptic locus is \((2g-1)\)-dimensional. Therefore, it is surprising that, in the tropical setting, it is \((3g-3)\)–dimensional. In this paper, the author proves that the locus of 2–edge–connected genus \(g\) tropical hyperelliptic curves is a \((2g-1)\)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on \(g-1\) vertices and maximum valence 3.
At the end of the paper, the author concentrates on the tropicalization of hyperelliptic curves \(X\) in the plane, over a complete non–archimedean valuated field \(K\). If the curve \(X\) is given by a degree 2 polynomial whose Newton polygon has vertices \((0,0), (2g+2,0)\) and \((0,2)\), the author proves that under certain combinatorial conditions, the Berkovich skeleton of \(X\) is a ladder over the path on \(g-1\) vertices.
The paper is carefully written and very informative.

MSC:

14T05 Tropical geometry (MSC2010)
14H10 Families, moduli of curves (algebraic)
14H51 Special divisors on curves (gonality, Brill-Noether theory)

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References:

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