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.

中文翻译：

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属六曲线、K3 曲面和稳定对

属 6 的一般平滑曲线位于 quintic del Pezzo 曲面上。在\cite{AK11} 中，Artebani 和Kond\=o 通过取del Pezzo 曲面的分枝双覆盖构造了属六曲线的双有理周期图。地图没有为特殊的属六曲线定义。在本文中，我们构建了一个平滑的 Deligne-Mumford 堆栈 $\mathfrak{P}_0$ 参数化某些稳定的表面曲线对，这基本上解决了这个映射。此外，我们给出了 $\mathfrak{P}_0$ 中包含特殊曲线的对的明确描述。