# Genus six curves, K3 surfaces, and stable pairs

## TL;DR

This paper constructs a moduli stack for stable surface-curve pairs to resolve a birational period map for genus six curves, including special cases, providing a comprehensive geometric framework.

## Contribution

It introduces a new moduli stack that parametrizes stable pairs, extending the period map to include special genus six curves, and offers explicit descriptions of these pairs.

## Key findings

- Constructed a smooth Deligne-Mumford stack for stable pairs.
- Resolved the indeterminacy of the period map for special curves.
- Provided explicit descriptions of pairs containing special genus six curves.

## Abstract

A general smooth curve of genus six lies on a quintic del Pezzo surface. In \cite{AK11}, Artebani and Kond\=o construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne-Mumford stack $\mathfrak{P}_0$ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in $\mathfrak{P}_0$ containing special curves.