Rational curves on holomorphic symplectic fourfolds

Details Rational curves on holomorphic symplectic fourfolds Rational curves on holomorphic symplectic fourfolds

1999-10-05

arxiv.org/abs/math/9910021v2

Geometric and Functional Analysis 11 (2001), no. 6, 1201-1228

Mathematics

Algebraic Geometry

33 pages, LaTeX; final version with Erratum included

Scientific paper

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral symmetric nondegenerate form, the Beauville form. We give precise conjectures for the structure of the cone of effective curves - and by duality - the cone of ample divisors. Formally they are completely analogous to the results known for K3 surfaces, and they are expressed entirely in terms of the integers represented by the Beauville form restricted to Pic(F). We prove that these conjectures are true in an open subset of the moduli space using deformation theory. The Fano variety of lines contained in a cubic fourfold is an example of a holomorphic symplectic fourfold. Our conjectures imply many concrete geometric statements about cubic fourfolds which may be verified using projective geometry.

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