MathDB

The effective curves conjecture for holomorphic symplectic fourfolds

From papers

Let (F,g)(F,g) be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to S[2]S^{[2]}, where SS is a K3 surface. Let N+1(F,g)\mathrm{N}^1_+(F,g) be the positive halfspace, and let EE be the set of classes ρ∈N+1(F,g)\rho\in\mathrm{N}^1_+(F,g) satisfying one of the three numerical conditions

(ρ,ρ)=−2, (ρ,H2(F,Z))=2Z;(\rho,\rho)=-2,\ (\rho,H^2(F,\mathbb Z))=2\mathbb Z; (ρ,ρ)=−2, (ρ,H2(F,Z))=Z;(\rho,\rho)=-2,\ (\rho,H^2(F,\mathbb Z))=\mathbb Z;

or

(ρ,ρ)=−10, (ρ,H2(F,Z))=2Z.(\rho,\rho)=-10,\ (\rho,H^2(F,\mathbb Z))=2\mathbb Z.

For the corresponding classes R∈H2(F,Z)R\in H_2(F,\mathbb Z), let NE(F,g)⊂H2(F,R)\mathrm{N}_E(F,g)\subset H_2(F,\mathbb R) be the smallest real cone containing E∗E^* and the classes R∈N1(F,Z)R\in\mathrm{N}_1(F,\mathbb Z) such that R.g>0R.g>0 and the corresponding ρ\rho has nonnegative square. Effective curves conjecture.

NE1(F)=NE(F,g).\mathrm{NE}_1(F)=\mathrm{N}_E(F,g).

This conjecture describes the cone of effective curves using the specified exceptional classes and classes of nonnegative square. Its status is not resolved in the supplied source context.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Effective curves conjecture for holomorphic symplectic fourfolds

    Let (F,g)(F,g) be a gg-polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let E∗E^* be the set of curve classes corresponding to divisor classes ρ∈Pic⁡+(F,g)\rho\in\operatorname{Pic}_+(F,g) with Beauville square −2-2 or −10-10 and the specified divisibility conditions, and let NE(F,g)⊂H2(F,Z)N_E(F,g)\subset H_2(F,\mathbb Z) be the smallest real cone containing E∗E^* and the classes R∈N1(F)R\in N_1(F) with R⋅g>0R\cdot g>0 whose corresponding ρ\rho has nonnegative square. Effective curves conjecture.

    NE(F)=NE(F,g).NE(F)=N_E(F,g).

    This predicts the full cone of effective curves from the distinguished negative-square classes and the remaining nonnegative-square classes. Its validity is the central conjectural step in the proposed description of the ample cone; the source gives no resolution.

    source: Brendan Hassett and Yuri Tschinkel, “Rational curves on holomorphic symplectic fourfolds”, arXiv:math/9910021 (2010).

Sources & referencesView supporting material

Primary source

Brendan Hassett and Yuri Tschinkel, “Moving and ample cones of holomorphic symplectic fourfolds”, arXiv:0710.0390 (2007).

cones-of-curveseffective-conesfourfoldsholomorphic-symplectic-geometryk3-surfaces

Solutions 0

No solutions have been posted yet.