Genus Six Curves, K3 Surfaces, and Stable Pairs

Abstract

Abstract A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondō [4] 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.