[Paper Review] A cascade of determinantal Calabi--Yau threefolds

Grzegorz Kapustka, Grzegorz Kapustka|ArXiv.org|Feb 25, 2008

Algebraic Geometry and Number Theory1 references5 citations

TL;DR

This paper constructs a cascade of determinantal Calabi–Yau threefolds via Kustin–Miller unprojections, linking families of Calabi–Yau threefolds defined by minors of generic matrices through conifold transitions. It computes their Hodge numbers and describes morphisms corresponding to faces of the Kähler–Mori cone, establishing a geometric analogy with del Pezzo surfaces via unprojection sequences.

ABSTRACT

We study Kustin--Miller unprojections of Calabi--Yau threefolds. As an application we work out the geometric properties of Calabi--Yau threefolds defined as linear sections of determinantal varieties. We compute their Hodge numbers and describe the morphisms corresponding to the faces of their Kähler--Mori cone.

Motivation & Objective

• To extend the class of explicitly constructible Calabi–Yau threefolds using determinantal ideals defined by minors of linear matrices.
• To establish a geometric cascade of Calabi–Yau threefolds analogous to Reid's cascades of del Pezzo surfaces, using Kustin–Miller unprojections.
• To compute Hodge numbers and describe the structure of the Kähler–Mori cone for specific families of determinantal Calabi–Yau threefolds.
• To provide a systematic method for constructing nodal Calabi–Yau threefolds containing Gorenstein subvarieties, applicable to unprojection theory.

Proposed method

• Utilizes Kustin–Miller unprojections to construct Calabi–Yau threefolds by resolving singularities of nodal varieties and contracting del Pezzo surfaces to points.
• Applies computer algebra (Singular) and Lemma 3.4 to verify nodal properties in specific examples, particularly for 3×3 minors of 4×4 matrices.
• Employs Grassmann blow-ups and results from determinantal varieties (e.g., from [CM], [Co], [JLP]) to analyze the geometry of the constructed threefolds.
• Uses Schubert calculus and Lascoux formulas to compute Chow groups and relations among Chern classes in the Chow ring of determinantal strata.
• Applies Namikawa’s results to compute Hodge numbers for threefolds defined by 3×3 minors of generic 4×4, 4×5 partially symmetric, and 5×5 symmetric matrices.
• Leverages an appendix by P. Pragacz on general determinantal varieties to support cohomological and Chow group computations.

Experimental results

Research questions

• RQ1 How can Kustin–Miller unprojections be used to construct a cascade of Calabi–Yau threefolds analogous to cascades of del Pezzo surfaces?
• RQ2 What are the Hodge numbers of Calabi–Yau threefolds defined as linear sections of determinantal varieties via 3×3 minors of generic matrices of linear forms?
• RQ3 Which morphisms correspond to the faces of the Kähler–Mori cone in these determinantal Calabi–Yau threefolds?
• RQ4 Can nodal Calabi–Yau threefolds of the specified determinantal type be constructed and resolved to yield smooth Calabi–Yau threefolds with desired geometric properties?
• RQ5 How do the Chow groups and relations among Chern classes behave on the strata of determinantal varieties arising in these constructions?

Key findings

• The Calabi–Yau threefold in ℙ⁷ defined by 3×3 minors of a generic 4×4 matrix of linear forms has Hodge numbers h¹¹ = 2 and h¹² = 34.
• The threefold in ℙ⁸ defined by 3×3 minors of a generic 4×5 partially symmetric matrix has Hodge numbers h¹¹ = 2 and h¹² = 25.
• The threefold in ℙ⁹ defined by 3×3 minors of a generic 5×5 symmetric matrix has Hodge numbers h¹¹ = 1 and h¹² = 26.
• The Kähler–Mori cone of each threefold admits a description via primitive contractions corresponding to its faces, which are explicitly analyzed.
• The Chow group A¹(Dᵣ) of the determinantal variety Dᵣ is isomorphic to ℤ⊕ℤ for generic 4×4 matrices and ℤ for symmetric cases, as shown via Schubert calculus and relations from Lascoux formulas.
• The Chow group Aₖ(Dᵣ∖Dᵣ₋₁) is generated over ℚ by Schur classes sⱼ(R) and powers of the hyperplane class h, with relations that allow reduction to finitely many generators.