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Tropical geometry of genus two curves. (English) Zbl 1408.14200
This article is about the structure of tropical genus 2 curves and their moduli.
There are various different methods to study tropical genus 2 curves. Some previous work in this direction can be found in [Q. Ren et al., Math. Comput. Sci. 8, No. 2, 119–145 (2014; Zbl 1305.14031)], [P. A. Helminck, “Tropical Igusa invariants and torsion embeddings”, arXiv:1604.03987].
The authors approach this problem by consideration of as metric graphs dual to genus 2 nodal algebraic curves over a valued field. From this point of view, they consider the moduli space of abstract genus two tropical curves, and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical framework. Igusa invariants were first defined in [J.-i. Igusa, Ann. Math. (2) 72, 612–649 (1960; Zbl 0122.39002)], in order to find an anology of the j-invariant to study genus 1 curves, for higher genus curves.
However, tropical Igusa functions do not yield coordinates on the tropical moduli space. The authors propose an alternative set of invariants that recovers length data.
MSC:
14T05 Tropical geometry (MSC2010)
14H45 Special algebraic curves and curves of low genus
14Q05 Computational aspects of algebraic curves
14G22 Rigid analytic geometry
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