[Paper Review] Fano threefolds of genus 6

Dmitry Logachev|ArXiv.org|Jul 8, 2004

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper investigates Fano threefolds of genus 6 as the intersection of the Grassmannian G(2,5) in P^9 with two hyperplanes and a quadric. Using the intermediate Jacobian and Abel-Jacobi map, it proves the Fano surface F(X) is smooth and irreducible, establishes a tangent bundle theorem with geometric interpretation, and shows F(X) uniquely determines X via a global Torelli-type result. The key contribution is proving the Abel-Jacobi map from the Albanese of F(X) to the middle Jacobian J^3(X) is an isogeny, with image algebraically equivalent to 2Θ^8 / 8!.

ABSTRACT

This paper was written in 1982. Ideas and methods of "Clemens C.H., Griffiths Ph. The intermediate Jacobian of a cubic threefold" are applied to a Fano threefold X of genus 6 -- intersection of Grassmann sixfold with two hyperplanes and a quadric. We prove: 1. The Fano surface F(X) of X is smooth and irreducible. Hodge numbers and some other invariants of F(X) are calculated. 2. Tangent bundle theorem for X, and its geometric interpretation. It is shown that F(X) defines X uniquely. 3. The Abel - Jacobi map from the Albanese of F(X) to the middle Jacobian of X is an isogeny.

Motivation & Objective

• To establish the smoothness and irreducibility of the Fano surface F(X) for a Fano threefold X of genus 6.
• To prove a tangent bundle theorem for X and provide its geometric interpretation.
• To demonstrate that the Fano surface F(X) uniquely determines X, establishing a global Torelli-type theorem.
• To investigate the Abel-Jacobi map Φ: Alb F(X) → J^3(X), proving it is an isogeny.
• To compute Hodge numbers and invariants of F(X), showing they match those of a double cover of P^3 branched in a quartic.

Proposed method

• Utilizes the intermediate Jacobian and Abel-Jacobi map to analyze the geometry of X and its Fano surface F(X).
• Applies Clemens and Griffiths' methods on the cubic threefold to the genus 6 Fano threefold X = G(2,5) ∩ P^7 ∩ Q, where Q is a quadric.
• Analyzes F(X) as a determinantal variety via Plücker embedding and studies its blow-up at a distinguished point c_Ω.
• Introduces an involution i_F on the blow-up F of F_c, leading to a two-sheeted covering p_F: F → F_0.
• Constructs a special intersection of three quadrics in P^6 with a double point, whose Hesse curve is a smooth plane sextic.
• Uses the structure of the bundle of quadrics on P(V^*) × P^5 and the map b: G_4 → P^4 to recover geometric data from F(X).

Experimental results

Research questions

• RQ1 Is the Fano surface F(X) of a genus 6 Fano threefold smooth and irreducible?
• RQ2 Does the tangent bundle theorem for X admit a geometric interpretation in terms of F(X)?
• RQ3 Can X be uniquely reconstructed from its Fano surface F(X), establishing a global Torelli theorem?
• RQ4 Is the Abel-Jacobi map Φ: Alb F(X) → J^3(X) an isogeny?
• RQ5 What is the algebraic equivalence class of the image of Φ in J^3(X), and how does it relate to the theta divisor Θ?

Key findings

• The Fano surface F(X) is smooth and irreducible, with Hodge numbers matching those of a double cover of P^3 branched in a quartic.
• The tangent bundle theorem for X is proven, and its geometric interpretation shows that F(X) determines X uniquely under certain conditions.
• The Abel-Jacobi map Φ: Alb F(X) → J^3(X) is an isogeny, confirming a deep link between the Albanese of F(X) and the middle Jacobian of X.
• The image of Φ is algebraically equivalent to 2Θ^8 / 8! in J^3(X), where Θ is a Poincaré divisor, a key result in the geometry of the intermediate Jacobian.
• A special intersection of three quadrics in P^6 with one double point is constructed, whose Hesse curve is a smooth plane sextic, crucial for computing h^{1,0}(F(X)).
• The Fano threefold X is uniquely recoverable from F(X) via the reconstruction of the Grassmannian G(2,5) and the linear system of quadrics containing X.