# Moduli spaces of curves in tropical varieties

## TL;DR

This paper develops a framework for constructing tropical moduli spaces of rational stable maps to tropical hypersurfaces, linking combinatorial data to algebraic geometric concepts of virtual fundamental classes.

## Contribution

It introduces a method to build tropical moduli spaces using combinatorial weights and compatibility conditions, applicable to lines in surfaces and beyond.

## Key findings

- Constructed tropical moduli spaces for lines in surfaces
- Provided a combinatorial framework ensuring well-defined tropical cycles
- Linked tropical moduli spaces to algebraic virtual fundamental classes

## Abstract

We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental classes in algebraic geometry. As we focus on the combinatorial aspect, we take the weights on certain basic 0-dimensional local combinatorial curve types as input data, and give a compatibility condition in dimension 1 to ensure that this input data glues to a global well-defined tropical cycle. As an application, we construct such moduli spaces for the case of lines in surfaces, and in a subsequent paper for stable maps to a curve.