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We study lines and twisted cubics on cubic fourfolds with simple isolatedsingularities. We show that the Hilbert scheme compactification of the totalspace of Starr's fibration on the space of twisted cubics on a cubic fourfoldwith simple isolated singularities and not containing a plane admits acontraction to a singular projective symplectic variety of dimension eight. Thelatter admits a crepant resolution which is a deformation of the symplecticeightfold constructed from the space of twisted cubics on a smooth cubicfourfold. Thereby we give another proof of the fact that the latter is adeformation of the Hilbert scheme of four points on a K3 surface. We also showthat the variety of lines on a cubic fourfold with simple singularities is asingular symplectic variety birational to the Hilbert scheme of two points on aK3 surface.

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