The moduli of curves of genus 6 and K3 surfaces

TL;DR

This paper establishes a birational equivalence between the moduli space of genus 6 curves and an arithmetic quotient of a type IV bounded symmetric domain, via a period map to lattice-polarized K3 surfaces.

Contribution

It introduces a novel period map linking genus 6 curves to lattice-polarized K3 surfaces, revealing the moduli space's structure as a quotient of a symmetric domain.

Findings

Moduli space of genus 6 curves is birational to an arithmetic quotient.

Constructs a period map to K3 surfaces.

Provides new insights into the geometry of genus 6 curves.

Abstract

We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.