Tropical hyperelliptic curves

By Melody Chan. 2011

88 citations recorded by the source corpus.

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove 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 with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/76ca914adabc12243ba5293ee26ccd7eec3d57b7

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Citations in the corpus

Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta (2013, unknown): “The definition we use is the same as the one given in [21,35].”
Moduli of tropical plane curves (2014, background): “M hyp g of all hyperelliptic metric graphs, seen in [12].”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, result): “The following proposition is a refinement of a result from [Chan 2013] on vertexweighted metric graphs (itself a strengthening of results from [Baker and Norine 2009]):”