The complex ball-quotient structure of the moduli space of certain sextic curves

Abstract

We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne–Mostow theory and periods of $K3$ surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of complex hyperbolic balls. We show in Theorem 7.4 that the two ball-quotient constructions can be unified in a geometric way.