Twisted cubics on a cubic fourfold

I will report on joint work (in progress) with C. Lehn,
C. Sorger and D. van Straten on a proposal for a
construction of a new family of 8-dimensional symplectic
manifold. Preliminary calculations of the Euler number
make it likely that the family parametrises deformation
of the Hilbertscheme Hilb^4 of a K3-Surface. The construction
involves classical geometry of singular cubic surfaces,
twisted cubic curves and geometric invariant theory
constructions.