Tropical geometry of genus two curves
Abstract
We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on the associated minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of P^1 with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2to1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding. Finally, we 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 realm. While these tropical Igusa functions do not yield coordinates in the tropical moduli space, we propose an alternative set of invariants that provides new length data.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1801.00378
 Bibcode:
 2018arXiv180100378A
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 14T05;
 14H45 (primary);
 14Q05;
 14G22 (secondary)
 EPrint:
 44 pages, 16 figures. Minor changes, including supplementary material