  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.
