Twisted cubics on singular cubic fourfolds - On Starr's fibration

Abstract

We study lines and twisted cubics on cubic fourfolds with simple isolated singularities. We show that the Hilbert scheme compactification of the total space of Starr's fibration on the space of twisted cubics on a cubic fourfold with simple isolated singularities and not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight. The latter admits a crepant resolution which is a deformation of the symplectic eightfold constructed from the space of twisted cubics on a smooth cubic fourfold. Thereby we give another proof of the fact that the latter is a deformation of the Hilbert scheme of four points on a K3 surface. We also show that the variety of lines on a cubic fourfold with simple singularities is a singular symplectic variety birational to the Hilbert scheme of two points on a K3 surface.