Klaus Hulek (Institute for Algebraic Geometry at Leibniz University of Hannover)

X

Cubic hypersurfaces are of special interest in geometry. The fact that every smooth cubic surface contains precisely 27 lines is one of the most classical results of algebraic geometry. Cubic hypersurfaces of dimension 3 became famous for anther reason: they were the first example known of a variety X which admits a finite dominant map from a projective space onto X (X is unirational), but is itself not birational to projective space (X is not rational). In dimension 4 we see yet another phenomenon: the lines on a cubic hypersurface of dimension 4 form an irreducible holomorphic symplectic (hyperkähler) manifold. In this talk I will discuss some new results on cubic hypersurfaces.