Moduli of tropical plane curves

Sarah Brodsky,Bernd Sturmfels,Michael Joswig,Ralph Morrison

doi:10.1186/s40687-014-0018-1

Abstract

Abstract We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus g, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with g interior lattice points. It has dimension 2g+1 unless g≤3 or g=7. We compute these spaces explicitly for g≤5.

Highlights

Tropical plane curves C are dual to regular subdivisions of their Newton polygon P
The curve C contains a subdivision of a metric graph of genus g with vertices of valency ≥ 3 as in [5], and this subdivision is unique for g ≥ 2
Scott [29] proved that # ∂P ∩ Z2 ≤ 2g + 7, and this bound is sharp. This means that the number of interior lattice points yields a bound on the total number of lattice points in P

Summary

Introduction

Tropical plane curves C are dual to regular subdivisions of their Newton polygon P. Let MP be the closure in Mg of the set of metric graphs that are realized by smooth tropical plane curves with Newton polygon P. For a fixed regular unimodular triangulation of P, let M be the closure of the cone of metric graphs from tropical curves dual to. We construct the stacky fans Mpglanar by computing each of the spaces MQ(ig) and subdividing their union appropriately This is augmented in Section ‘Hyperelliptic curves’ by the spaces MP where Pint is not 2-dimensional, but is instead a line segment. We summarize the objects discussed so far in a diagram of surjections and inclusions: MP ⊆

Mpg lanar

Td form a projective space of dimension

Findings

Genus three