Moduli spaces of sextic curves with simple singularities and their compactifications

TL;DR

This paper investigates the structure and compactifications of moduli spaces of sextic curves with simple singularities, using period maps of K3 surfaces to establish algebraic embeddings and identify compactifications.

Contribution

It provides new results on the algebraic open embeddings of these moduli spaces into arithmetic quotients and explicitly describes Picard lattices for various cases.

Findings

Moduli spaces admit algebraic open embeddings into arithmetic quotients.

GIT and Looijenga compactifications are identified for all cases.

Orbifold structures are isomorphic in nodal cases.

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.