[Paper Review] Intersection numbers of extremal rays on holomorphic symplectic varieties

Brendan Hassett, Yuri Tschinkel|arXiv (Cornell University)|Sep 25, 2009

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper proposes a universal rational constant $ c_X $ governing the intersection numbers of extremal rays on irreducible holomorphic symplectic manifolds, generalizing the K3 surface case. It establishes that the pseudoeffective cone of curves $ \overline{\mathrm{NE}}_1(X) $ is generated by classes $ R \in \mathrm{N}_1(X,\mathbb{Z}) $ with $ (R,R) \geq -c_X $ and positive intersection with a polarization, proving this for punctual Hilbert schemes $ S^{[n]} $ and Kummer varieties $ K_n(A) $, where $ c_X = (n+3)/2 $ and $ c_X = 2/(g+1) $, respectively.

ABSTRACT

We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.

Motivation & Objective

• To extend the explicit description of the ample and pseudoeffective cones on K3 surfaces to higher-dimensional irreducible holomorphic symplectic manifolds.
• To identify a universal rational constant $ c_X $, depending only on the deformation type of $ X $, that bounds the self-intersection of extremal curve classes.
• To characterize the cone of effective curves $ \mathrm{NE}_1(X) $ via intersection-theoretic conditions: $ (R,R) \geq -c_X $ and $ R \cdot g > 0 $ for a polarization $ g $.
• To verify the conjecture for specific deformation types: punctual Hilbert schemes $ S^{[n]} $, Kummer varieties $ K_n(A) $, and Lagrangian $ \mathbb{P}^n $-fibrations.
• To relate extremal ray geometry to birational contractions, particularly divisorial contractions, and to determine the structure of effective divisor cones.

Proposed method

• Use the Beauville-Bogomolov form on $ H^2(X,\mathbb{Z}) $ to define a $ \mathbb{Q} $-valued intersection form on $ H_2(X,\mathbb{Z}) $, which governs curve and divisor intersection numbers.
• For Hilbert schemes $ S^{[n]} $, decompose $ H^2(X,\mathbb{Z}) $ as $ H^2(S,\mathbb{Z}) \oplus_\perp \mathbb{Z}\delta $, with $ (\delta,\delta) = -2(n-1) $, and dualize to $ \delta^\vee $ with $ (\delta^\vee,\delta^\vee) = -1/2(n-1) $.
• For Kummer varieties $ K_n(A) $, use the Néron-Severi group generated by $ \Theta $ and $ e $, with $ (\Theta,\Theta) = 2g-2 $, $ (e,e) = -2(g+1) $, and analyze rulings on fibers of relative Jacobians.
• Construct extremal rays $ R $ as classes of rulings on $ \mathbb{P}^k $-bundles over moduli spaces of line bundles on curves, computing $ (R,R) $ via intersection with $ \Theta $ and $ e $.
• Define $ \rho = kR $ as the smallest positive multiple in $ \mathrm{H}^2(X,\mathbb{Z}) $, and verify $ (\rho,\rho) = (e,e) $, linking to the universal constant $ c_X $.
• Use deformation theory and moduli of curves to construct families of subschemes $ Z \subset A $ of fixed length, fibered over Jacobians, to realize extremal rays as fibers of $ \mathbb{P}^1 $-bundles.

Experimental results

Research questions

• RQ1 What is the universal rational constant $ c_X $ such that all extremal curve classes $ R \in \overline{\mathrm{NE}}_1(X) $ satisfy $ (R,R) \geq -c_X $ for a holomorphic symplectic manifold $ X $?
• RQ2 How can the pseudoeffective cone $ \overline{\mathrm{NE}}_1(X) $ be described using intersection-theoretic conditions involving a polarization $ g $?
• RQ3 Does the conjecture $ \mathrm{NE}_1(X) = \langle R \in \mathrm{N}_1(X,\mathbb{Z}) : (R,R) \geq -c_X, R \cdot g > 0 \rangle $ hold for deformation types like $ S^{[n]} $ and $ K_n(A) $?
• RQ4 What is the geometric meaning of extremal rays with negative self-intersection, and how do they relate to birational contractions?
• RQ5 Can the intersection number $ (R,R) $ of an extremal ray $ R $ be computed explicitly from the geometry of the associated contraction?

Key findings

• For $ X $ deformation equivalent to $ S^{[n]} $, the pseudoeffective cone of curves is $ \mathrm{NE}_1(X) = \langle R \in \mathrm{N}_1(X,\mathbb{Z}) : (R,R) \geq -(n+3)/2, R \cdot g > 0 \rangle $, with $ c_X = (n+3)/2 $.
• For $ X = K_n(A) $, the constant $ c_X = 2/(g+1) $, and the extremal ray $ R $ from a $ \mathbb{P}^1 $-bundle over a Jacobian satisfies $ (R,R) = -2/(g+1) $, matching $ c_X $.
• In the case of $ K_n(A) $ with $ (1,g) $-polarization, the extremal ray $ R = \Theta - \frac{g}{g+1}e $ has $ (R,R) = -2/(g+1) $, and $ \rho = (g+1)R $ satisfies $ (\rho,\rho) = -2(g+1) = (e,e) $.
• For $ X = K_n(A) $ with $ A = E_1 \times E_2 $, the line $ \ell \subset \mathbb{P}^n $-fibration satisfies $ R = E_1 - \frac{1}{2}e $, with $ (R,R) = -(n+1)/2 $, so $ c_X = (n+1)/2 $, and $ \rho = 2R $ gives $ (\rho,\rho) = -2(n+1) = (e,e) $.
• In all studied cases, the smallest positive multiple $ \rho $ of the extremal ray $ R $ satisfies $ (\rho,\rho) = (e,e) $, supporting the universality of the constant $ c_X $.
• The extremal ray $ R $ from a $ \mathbb{P}^1 $-bundle over a $ (2g-2) $-dimensional moduli space satisfies $ R = \Theta - \frac{g}{g+1}e $, and $ (R,R) = -2/(g+1) $, confirming the conjecture for $ K_g(A) $.