Compactifications of the moduli space of plane quartics and two lines

摘要

We study the moduli space of triples $(C, L_1, L_2)$ consisting of quarticcurves $C$ and lines $L_1$ and $L_2$. Specifically, we construct and compactifythe moduli space in two ways: via geometric invariant theory (GIT) and by usingthe period map of certain lattice polarized $K3$ surfaces. The GIT constructiondepends on two parameters $t_1$ and $t_2$ which correspond to the choice of alinearization. For $t_1=t_2=1$ we describe the GIT moduli explicitly and relateit to the construction via $K3$ surfaces.