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

Radu Laza; Gregory Pearlstein; Zheng Zhang

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arXiv · 1710.08056

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

Radu Laza·Gregory Pearlstein·Zheng Zhang

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Abstract

We study the moduli space of pairs $(X,H)$ consisting of a cubic threefold $X$ and a hyperplane $H$ in $\mathbb P^4$. 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 $O_{16}$ (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.

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Radu Laza, Gregory Pearlstein, Zheng Zhang. 2017-10-23. On the moduli space of pairs consisting of a cubic threefold and a hyperplane. https://arxiv.org/abs/1710.08056

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