Moduli spaces of sextic curves with simple singularities and their compactifications
Chenglong Yu, Zhiwei Zheng, Yiming Zhong
TL;DR
The paper develops a period-map framework linking moduli spaces of sextic plane curves with ADE singularities to arithmetic quotients of type IV domains via K3 surfaces arising from double covers. It proves that the occult period maps yield open embeddings whose images are the complements of Heegner-type hyperplanes, and identifies the GIT compactifications with Looijenga compactifications for many singular types. Through explicit lattice descriptions and topological lemmas, it provides conditions under which the generic Picard lattice and related invariants can be computed, and demonstrates when the period map preserves orbifold structures, notably in nodal cases. The work includes detailed examples (quintics, quartics with bitangents) and connects to broader moduli phenomena such as Zariski pairs, illustrating the rich interaction between algebraic geometry, lattice theory, and arithmetic groups in the study of sextic curves.
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.