Thesis advisor: Maksym Fedorchuk / A general smooth curve of genus six lies on a quintic del Pezzo surface. In [AK11], Artebani and Kondō 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 dissertation, we construct a smooth Deligne-Mumford stack P₀ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in P₀ containing special curves. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.