[Paper Review] On the moduli space of pairs consisting of a cubic threefold and a hyperplane

Radu Laza, Gregory Pearlstein+1 more|arXiv (Cornell University)|Oct 23, 2017

Algebraic Geometry and Number Theory30 references3 citations

TL;DR

This paper constructs a Hodge-theoretic period map for the moduli space of pairs (X,H), where X is a cubic threefold and H is a hyperplane in ℙ⁴, by associating each such pair to a lattice-polarized cubic fourfold Y with an Eckardt point. The main result is that the period map induces an isomorphism between a GIT quotient model of the moduli space and the Baily-Borel compactification of a type IV locally symmetric domain.

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.

Motivation & Objective

• To construct a Hodge-theoretic moduli space for pairs (X,H) consisting of a cubic threefold X and a hyperplane H in ℙ⁴.
• To relate such pairs to cubic fourfolds Y that admit an Eckardt point, i.e., contain a cone over a cubic surface as a hyperplane section.
• To define a period map for the pairs (X,H) using the periods of the associated cubic fourfolds Y.
• To establish that the period map induces an isomorphism between the GIT quotient model of the moduli space and the Baily-Borel compactification of a type IV locally symmetric domain.
• To provide a Hodge-theoretic compactification of the period map via lattice polarization and Torelli-type theorems.

Proposed method

• Construct a cubic fourfold Y from a pair (X,H) by taking the double cover of ℙ⁴ branched along X ∪ H, which yields a quasi-smooth weighted hypersurface of degree 6 in ℙ(1,2,2,2,2,3).
• Characterize the associated cubic fourfolds Y as M-polarized, where M is a saturated sublattice of H⁴(Y,ℤ) ∩ H²²(Y) generated by the hyperplane class and a scaled E₆ lattice from the cubic surface X ∩ H.
• Use Voisin’s Torelli theorem for cubic fourfolds and lattice theory to show that the moduli space of (X,H) is birational to a locally symmetric domain of type IV.
• Define the period map 𝒫₀: 𝒪₀ → 𝒟ₘ / O⁺(T), where T = M⊥ in H⁴(Y,ℤ), and 𝒟ₘ is the period domain for weight 4 Hodge structures on T with Hodge numbers (0,1,14,1,0).
• Prove that the period map is an isomorphism onto its image by leveraging the Torelli theorem and the geometry of the lattice M.
• Compactify the period map by extending the construction to the Baily-Borel compactification of the locally symmetric domain.

Experimental results

Research questions

• RQ1 How can the moduli space of pairs (X,H) consisting of a cubic threefold and a hyperplane be constructed using Hodge theory?
• RQ2 What is the Hodge-theoretic characterization of cubic fourfolds arising from such pairs (X,H)?
• RQ3 Can a period map for the moduli space of (X,H) be defined via the periods of associated cubic fourfolds Y?
• RQ4 Is the period map from the moduli space of (X,H) to a locally symmetric domain an isomorphism?
• RQ5 How does the compactification of the period map relate to the Baily-Borel compactification of the symmetric domain?

Key findings

• The moduli space of smooth cubic pairs (X,H) is birational to a locally symmetric domain of type IV.
• The period map 𝒫₀: 𝒪₀ → 𝒟ₘ / O⁺(T) is an isomorphism onto its image, establishing a Hodge-theoretic uniformization of the moduli space.
• The associated cubic fourfolds Y are characterized as M-polarized, where M is a saturated sublattice of H⁴(Y,ℤ) ∩ H²²(Y) containing a scaled E₆ lattice from the cubic surface X ∩ H.
• The period domain 𝒟ₘ parametrizes weight 4 Hodge structures on the orthogonal complement T = M⊥ with Hodge numbers (0,1,14,1,0).
• The group O⁺(T) acts as the stabilizer of 𝒟ₘ, and the quotient 𝒟ₘ / O⁺(T) is isomorphic to the GIT quotient model of the moduli space of (X,H).
• The construction provides a Hodge-theoretic compactification of the moduli space via the Baily-Borel compactification of the symmetric domain.