Compactifications of the moduli space of plane quartics and two lines

Abstract

We study the moduli space of triples (C, L1, L2) consisting of quartic curves C and lines L1 and L2. 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 t1 and t2 which correspond to the choice of a linearization. For t1= t2= 1 we describe the GIT moduli explicitly and relate it to the construction via K3 surfaces.