# Rational curves in the logarithmic multiplicative group

## TL;DR

This paper investigates the structure of moduli stacks of rational logarithmic maps to the logarithmic multiplicative group, revealing they often decompose into a product of the group and rational curves, clarifying previous results.

## Contribution

It provides a structural description of the stacks of logarithmic maps to the logarithmic torus, explaining their product structure and connecting to earlier work on toric varieties.

## Key findings

- Stacks of logarithmic maps often decompose into a product of the logarithmic torus and rational curves.
- The results offer a conceptual understanding of moduli spaces of logarithmic stable maps to toric varieties.
- The study clarifies the structure of these stacks in most cases.

## Abstract

The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. We study the structure of the stacks of logarithmic maps from rational curves to this logarithmic torus, and show that in most cases, it is a product of the logarithmic torus with the space of rational curves. This gives a conceptual explanation for earlier results on the moduli spaces of logarithmic stable maps to toric varieties.