[Paper Review] Calabi-Yau Threefolds and Moduli of Abelian Surfaces I

Mark Gross, Sorin Popescu|ArXiv.org|Jan 14, 2000

Algebraic Geometry and Number Theory18 citations

TL;DR

This paper constructs birational models for moduli spaces of polarized abelian surfaces with canonical level structure, specifically proving that Alev_6 is birational to a smooth quadric in P^4, Alev_8 is birational to a rational conic bundle over P^2, and Alev_10 is birational to a quotient P^3/Z2×Z2. These results establish rationality or unirationality for these moduli spaces and reveal deep connections to Calabi-Yau threefolds via abelian surface fibrations on resolutions of singular threefolds defined by symmetric matrices.

ABSTRACT

We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective lines identified in the rational/unirational moduli spaces correspond to pencils of abelian surfaces traced on nodal threefolds living naturally in the corresponding ambient projective spaces, and whose small resolutions are new Calabi-Yau threefolds with Euler characteristic zero.

Motivation & Objective

• To determine the birational geometry of moduli spaces Alev_d for small d, especially d = 6, 8, 10.
• To establish rationality or unirationality of these moduli spaces by constructing explicit birational models.
• To explore the role of Calabi-Yau threefolds arising as resolutions of singular threefolds containing pencils of (1,d)-polarized abelian surfaces.
• To connect the geometry of these moduli spaces to known examples such as the Horrocks-Mumford quintic and prime Fano threefolds.
• To provide a foundation for determining Kodaira dimension and understanding uniruledness in higher-dimensional moduli spaces.

Proposed method

• Use of symmetric matrices and linear systems to define threefolds X ⊂ P^7 (for d=8) and X ⊂ P^4 (for d=6), which contain the abelian surfaces as complete intersections.
• Construction of small resolutions π: eX → X to produce Calabi-Yau threefolds with abelian surface fibrations.
• Analysis of the homogeneous ideal of (1,d)-polarized abelian surfaces, showing that for d=8, the ideal is generated by 4 quadrics and 16 cubics.
• Use of Heisenberg invariance and moduli of symmetric matrices to show that the family of such threefolds forms a P^2-bundle, leading to the conic bundle structure for Alev_8.
• Explicit birational parametrization of Alev_10 via quotient of P^3 by Z2×Z2, using symmetric matrix constructions.
• Comparison of resolutions of singular threefolds with Horrocks-Mumford quintics to deduce Calabi-Yau structure and singularity type (50 ordinary double points).

Experimental results

Research questions

• RQ1 Is Alev_6 birational to a smooth quadric in P^4, and what does this imply for its rationality?
• RQ2 Can Alev_8 be described as a rational conic bundle over P^2, and how does this arise from the geometry of its defining threefold?
• RQ3 What is the birational structure of Alev_10, and how does the Z2×Z2 quotient of P^3 relate to its moduli space?
• RQ4 How do the Calabi-Yau threefolds constructed from symmetric matrices relate to known examples like the Horrocks-Mumford quintic?
• RQ5 What is the Kähler cone structure of the minimal models of these Calabi-Yau threefolds, and how does it reflect the geometry of the fibrations?

Key findings

• Alev_6 is birational to a non-singular quadric hypersurface in P^4, establishing its rationality.
• Alev_8 is birational to a rational conic bundle over P^2, arising from a P^2-family of Heisenberg-invariant threefolds with 64 ordinary double points.
• Alev_10 is birational to the quotient P^3/Z2×Z2, where the action is given by (x1:x2:x3:x4) ↦ (x4:x3:x2:x1) and (x1:x2:x3:x4) ↦ (x1:−x2:x3:−x4), and this quotient is rational.
• The threefold X ⊂ P^7 for d=8 is a complete intersection of 4 quadrics and 16 cubics, with singular locus of 64 ordinary double points.
• The small resolution of X yields a Calabi-Yau threefold with an abelian surface fibration, and this resolution is isomorphic to a double cover of a Horrocks-Mumford quintic.
• For general y, the threefold W10,y is a Calabi-Yau threefold with 25 ordinary double points, obtained as an unbranched double cover of a singular threefold defined by symmetric matrices.