On the moduli space of pairs consisting of a cubic threefold and a hyperplane

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We study the moduli space of pairs (X,H) consisting of a cubic threefold X and a hyperplane H in P4. The interest in this moduli comes from two sources: the study of certain weighted hypersurfaces whose middle cohomology admit Hodge structures of K3 type and, on the other hand, the study of the singularity O16 (the cone over a cubic surface). In this paper, we give a Hodge theoretic construction of the moduli space of cubic pairs by relating (X,H) to certain “lattice polarized” cubic fourfolds Y. A period map for the pairs (X,H) is then defined using the periods of the cubic fourfolds Y. The main result is that the period map induces an isomorphism between a GIT model for the pairs (X,H) and the Baily–Borel compactification of some locally symmetric domain of type IV.