### 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.

Original language | English |
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Pages (from-to) | 1000-1034 |

Number of pages | 35 |

Journal | European Journal of Mathematics |

Volume | 4 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Sep 2018 |

### Keywords

- K3 surfaces
- variations of GIT quotients
- period map
- quartic curves

### ASJC Scopus subject areas

- Geometry and Topology
- Algebra and Number Theory

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## Cite this

Martinez-Garcia, J., Gallardo, P., & Zhang, Z. (2018). Compactifications of the moduli space of plane quartics and two lines.

*European Journal of Mathematics*,*4*(3), 1000-1034. https://doi.org/10.1007/s40879-018-0248-7