Cubic hypersurfaces, their Fano schemes, and special subvarieties

Abstract

This thesis investigates cubic hypersurfaces and their Fano schemes. After introducing the Fano schemes through low-dimensional examples, we move on to investigate cubic fourfolds. For the cubic fourfolds, we give complete proofs of some statements of Beauville-Donagi and Amerik. Following Beauville-Donagi, we introduce the Abel-Jacobi map. Together with its transpose, we use this to investigate cubic fourfolds and their varieties of lines. This is used to -among other things- to prove the Hodge conjecture for cubic fourfolds. For some cubics, we are able to prove the Integral Hodge conjecture. We also investigate linear subspaces of varieties. Here we generalize the techniques of Clemens and Griffiths, which leads to characterizations of linear spaces tangent to hypersurfaces. We continue by investigating the Eckardt points on cubic hypersurfaces. Studying these points is not a new idea, but our approach focusing on the lines through an Eckardt point is, -as far as we know- novel. We give several other characterizations of these points, and show that they influence whether a cubic fourfold is rational. Following this, we investigate some highly special cubic fourfolds, such as the Fermat cubic. We prove the Hodge conjecture for their Fano schemes. The second-to-last chapter introduces special cubic fourfolds, following the classification of Hassett . We describe some of the divisors in the moduli space of cubic fourfolds explicitly. These investigations lead us to answer a question raised by Nuer on the existence of smooth rational surfaces in cubic fourfolds. The chapter continues by discussing the effective and nef cones of 2-cycles on special cubic fourfolds. We give a new and complete description of their cones for fourfolds containing a plane. Some conjectures of Hassett and Tschinkel lead us to investigate the cones of nef cycles on their Fano schemes, and we fill in some details of their paper. The final chapter deviates from the theme. It is a vast generalization of our analysis of cubic fourfolds containing a plane. We give a complete description of the cones of effective and nef m-cycles for hypersurfaces of dimension 2m of sufficiently large degree. In certain cases, toric geometry leads to improved results. This result is, as far as we know, new