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

TL;DR

This paper investigates the geometry of lines and twisted cubics on singular cubic fourfolds, revealing new symplectic structures and deformations related to K3 surfaces, expanding understanding of their moduli spaces.

Contribution

It constructs a contraction of Starr's fibration on singular cubic fourfolds, showing it leads to a singular symplectic variety with a crepant resolution related to smooth cases.

Findings

Hilbert scheme compactification admits contraction to a singular symplectic variety

The singular symplectic variety admits a crepant resolution as a deformation of the smooth case

The variety of lines on a singular cubic fourfold is birational to a Hilbert scheme of two points on a K3 surface

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.