Twisted cubics on singular cubic fourfolds—On Starr’s fibration

Abstract

We show that the Hilbert scheme compactification of the total space of Starr’s fibration on the space of twisted cubics on a cubic hypersurface in $${\mathbb P}^5$$ not containing a plane admits a contraction to a singular projective symplectic variety of dimension eight which has a crepant resolution deformation equivalent to the symplectic eightfold constructed from twisted cubics on a smooth cubic fourfold. This yields another proof that the symplectic eightfold and the Hilbert scheme of four points on a K3 surface are deformation equivalent. As a byproduct we obtain similar results for the variety of lines on a singular cubic fourfold.