On the symplectic eightfold associated to a Pfaffian cubic fourfold

Abstract

Abstract We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z → Hilb 4 ⁡ ( X ) Z\to\operatorname{Hilb}^{4}(X) , where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points. This note is an appendix to http://dx.doi.org/10.1515/crelle-2014-0144.