Moduli spaces of sextic curves with simple singularities and their compactifications

Abstract

In this paper, we study moduli spaces of sextic curves with simple singularities. Through period maps of K3 surfaces with ADE singularities, we prove that such moduli spaces admit algebraic open embeddings into arithmetic quotients of type IV domains. For all cases, we prove the identifications of GIT compactifications and Looijenga compactifications. We also describe Picard lattices in an explicit way for many cases. For nodal cases, we prove that the orbifold structures on the two sides of the period map are isomorphic.