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 its minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of the projective line with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 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.

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves

“…As an application, they are able to solve the realisability problem for tropical maps of genus one (see Speyer [69]): which of them arise as the tropicalisation of a map from a smooth genus one curve to an algebraic torus? With Ranganathan, we are working towards a similar result in genus two -where, as far as we know, there is at the moment no reasonable guess as to what the full answer should be, although a clear understanding of the moduli space of tropical curves has been obtained by Cueto and Markwig [27].…”