Abstract

In this thesis we study deformations of varieties of lines on smooth cubic hypersurfaces of the 5-dimensional complex projective space. These cubic hypersurfaces are also called cubic fourfolds. Beauville and Donagi proved that the line variety of any cubic fourfold is a polarized holomorphic symplectic fourfold of K3[2]-type, with polarization
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class of degree 6 and of even intersection with the second cohomology lattice, with respect to the Beauville-Bogomolov form. Furthermore, they showed that for such line varieties any small complex deformation for which the polarization class remains algebraic is again the line variety of a cubic fourfold. However, it turns out that these line varieties can be deformed, while keeping the polarization class algebraic, in such a way that the resulting manifold is no longer isomorphic to the line variety of any cubic fourfold. Let us call such deformations special. We show, based on the work of in the thesis of B. Hassett, that every such special deformation is isomorphic to the Hilbert scheme of length 2 subschemes of a K3 surface of degree 2 or 6, and we identify the line bundle which is the deformation of the polarization. Also, we explain in terms of the properties of their holomorphic symplectic structure why these special deformations cannot be the line variety of any cubic fourfold. It follows from results by Hassett, Laza and Looijenga that these special deformations are related to certain singular cubic hypersurfaces, namely cubic hypersurfaces with an ordinary double point and the so-called determinantal cubic. Although a special deformation cannot be obtained as line variety of a single cubic fourfold, we show that in some cases it can be projectively reconstructed from a family of cubic fourfolds that degenerates to one of the singular cubic hypersurfaces mentioned earlier. We identify those special deformations, we describe the reconstruction explicitly and we show why the other special deformations cannot be reconstructed in a similar way. Finally, we show how our results give us precise and explicit characterizations for a line bundle of degree 6 and of even intersection with the second cohomology lattice (with respect to the Beauville-Bogomolov form) on a holomorphic symplectic manifold of K3[2]-type to be very ample, ample and/or nef. To our knowledge this is the first instance of such characterizations of (very-)ampleness of line bundles of a given numerical type on holomorphic symplectic manifolds of dimension higher than 2
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