[Paper Review] Theta constants associated to cubic three folds

Keiji Matsumoto, Tomohide Terasoma|ArXiv.org|Aug 3, 2000

Algebraic Geometry and Number Theory5 references4 citations

TL;DR

This paper establishes a W(E₆)-equivariant projective embedding of the moduli space of cubic surfaces with 27 lines using theta constants associated to the (1−ρ)-torsion points of the intermediate Jacobian of a triple cover of P³ branched over a cubic surface. The key result is that this embedding, derived from theta constants on the 4-dimensional complex ball, coincides with the classical projective embedding of the moduli space of six points in general position in P², thereby providing an explicit inverse to the period map via theta constants.

ABSTRACT

For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points.

Motivation & Objective

• To construct a W(E₆)-equivariant projective embedding of the moduli space of cubic surfaces with marked 27 lines using theta constants.
• To establish a precise correspondence between theta constants on the 4-dimensional complex ball and the classical coordinates of the moduli space of six points in P².
• To prove that the inverse of the period map from the moduli space of cubic surfaces to the ball quotient is given explicitly by these theta constants.
• To understand the action of the Weyl group W(E₆) on the (1−ρ)-torsion subgroup of the intermediate Jacobian of a triple cover of P³ over a cubic surface.

Proposed method

• Define theta constants Θᵥ(τ) on the 4-dimensional complex ball B₄ associated to elements v in the (1−ρ)-torsion subgroup of the intermediate Jacobian J(Y).
• Use the isomorphism between the Prym variety of a μ₃-covering of a line in the cubic surface and J(Y) to relate periods of J(Y) to those of a genus 11 curve.
• Leverage the action of the Weyl group W(E₆) on the 27 lines and the associated F₃-vector space structure on J(Y)₁₋ρ to classify the 80 non-vanishing theta constants.
• Construct a projective embedding Θ: Γ\B₄ → P⁷⁹ using the cubes of the 80 non-vanishing theta constants, which is W(E₆)-equivariant.
• Prove that this embedding coincides with the classical W(E₆)-equivariant embedding Z of the moduli space M₆pts of six points in P² via a commutative diagram involving the period map φ.
• Use combinatorial identities and Plücker-type relations in F₃⁵ to derive cubic relations among the theta constants and Z-polynomials.

Experimental results

Research questions

• RQ1 How can the inverse of the period map from the moduli space of cubic surfaces to the ball quotient Γ\B₄ be explicitly described?
• RQ2 What is the relationship between the W(E₆)-equivariant projective embedding of the moduli space of six points in P² and the theta constants on the ball?
• RQ3 How does the Prym construction of the intermediate Jacobian relate to the period map and theta constants?
• RQ4 What are the algebraic relations satisfied by the theta constants associated to the (1−ρ)-torsion points?
• RQ5 Can the classical embedding of the moduli space of six points in P² be recovered from the theta constants on the ball?

Key findings

• The projective embedding Θ of Γ\B₄ using the cubes of the 80 non-vanishing theta constants is W(E₆)-equivariant and coincides with the classical embedding Z of the moduli space M₆pts of six points in P².
• The inverse of the period map φ: M_cs → Γ\B₄ is explicitly given by the theta constants, with the map being an isomorphism onto its image.
• The 80 non-vanishing theta constants correspond bijectively to the 80 isotropic vectors in F₃⁵ under the quadratic form q, and their cubes define the embedding.
• The period map is proper and surjective, and its injectivity implies that the period map is an isomorphism, confirming the uniformization of the moduli space.
• Cubic relations among the theta constants are derived, including ∑_{v∈Ĩ(w)} Θᵥ³ = 0 for w ∈ R, and degree-9 identities ∏_{v₁∈S_{V₁}} Θᵥ₁³ = ε̃(S_{V₁},S_{V₂}) ∏_{v₂∈S_{V₂}} Θᵥ₂³ for maximal isotropic subspaces V₁,V₂ with one-dimensional intersection.
• The system of equations ∑_{v∈Ĩ(w)} Zᵥ = 0 and ∏_{v₁∈S_{V₁}} Zᵥ₁ = ε̃(S_{V₁},S_{V₂}) ∏_{v₂∈S_{V₂}} Zᵥ₂ defines the closure of M₆pts in P⁷⁹.