Moduli spaces of cubic hypersurfaces through a period map

Abstract

There are several approaches to studying moduli spaces; the most well-known in algebraic geometry is probably "Geometric Invariant Theory". In practice, the calculations involved quickly grow out of control. The study of K3 surfaces suggested a different method: a K3 surface is determined by its complex structure (or period), a result known as the Torelli theorem. Thus the study of their periods translates back to the parameter space of K3 surfaces. Similar results hold for more general period maps, which are studied. For cubic fourfolds there exists a period map similar to that of a K3 surface---that is, an injective map into an open part of a symmetric domain, having signature (2,n)---we can use the "Baily Borel" compactification of the image to describe a geometrically meaningful compactification of their moduli space. We use this map and the theory developed in earlier chapters to describe the moduli of cubic surfaces and threefolds.