Compactifications of the moduli space of plane quartics and two lines

TL;DR

This paper explores the compactification of the moduli space of plane quartics with two lines using GIT and $K3$ surface period maps, providing explicit descriptions and relations between the two approaches.

Contribution

It introduces two methods for compactifying the moduli space of quartic curves with lines and relates GIT stability to $K3$ surface periods.

Findings

Explicit GIT compactification for specific parameters

Relation between GIT stability and $K3$ period map

Description of the moduli space structure

Abstract

We study the moduli space of triples $(C, L_1, L_2)$ consisting of quartic curves $C$ and lines $L_1$ and $L_2$. Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized $K3$ surfaces. The GIT construction depends on two parameters $t_1$ and $t_2$ which correspond to the choice of a linearization. For $t_1=t_2=1$ we describe the GIT moduli explicitly and relate it to the construction via $K3$ surfaces.