Skeletons of stable maps II: Superabundant geometries

TL;DR

This paper develops new techniques using Artin fans to analyze the realizability of superabundant tropical stable maps, leading to key theorems on realizability, extension of well-spacedness sufficiency, and existence of counterexamples.

Contribution

It introduces a novel approach involving Artin fans to study the realizability of superabundant tropical maps, extending existing conditions and providing new counterexamples.

Findings

Proved a realizability theorem for limits of tropical stable maps.

Extended the sufficiency of Speyer's well-spacedness condition.

Constructed genus 1 superabundant tropical curves that violate well-spacedness.

Abstract

We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive 3 principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition.