# The GIT moduli of semistable pairs consisting of a cubic curve and a line on ${\mathbb P}^{2}$

## TL;DR

This paper analyzes the GIT moduli space of pairs consisting of a cubic curve and a line on the projective plane, comparing it with other moduli spaces and compactifications related to elliptic curves.

## Contribution

It provides a detailed study of the GIT moduli of semistable pairs and compares it with Alexeev's moduli and Nakamura's compactification, offering new insights into their relationships.

## Key findings

- Characterization of the GIT moduli space of pairs
- Comparison with Alexeev's moduli of pairs
- Analysis of Nakamura's compactification of elliptic curves

## Abstract

We discuss the GIT moduli of semistable pairs consisting of a cubic curve and a line on the projective plane. We study in some detail this moduli and compare it with another moduli suggested by Alexeev. It is the moduli of pairs (with no specified semi-abelian action) consisting of a cubic curve with at worst nodal singularities and a line which does not pass through singular points of the cubic curve. Meanwhile, we make a comparison between Nakamura's compactification of the moduli of level three elliptic curves and these two moduli spaces.