The Complex Ball-quotient Structure of the Moduli Space of Certain Sextic Curves

TL;DR

This paper explores the structure of moduli spaces of specific sextic curves with a triple point, demonstrating their description as complex hyperbolic ball quotients through two different theoretical approaches and unifying these perspectives.

Contribution

It unifies two distinct descriptions of the moduli space of certain sextic curves as complex hyperbolic ball quotients, linking Deligne-Mostow theory and K3 surface periods.

Findings

Moduli spaces are described as arithmetic quotients of complex hyperbolic balls.

The two different constructions of these moduli spaces are shown to be geometrically equivalent.

A unification of the two approaches provides a comprehensive understanding of the moduli space structure.

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.