Conics on projective hypersurfaces

Abstract

We expect that a projective hypersurface of smaller degree can contain more rational curves. Let X be an n-dimensional projective hypersurface of degree d. Assume that X is of Fano type, that is, $d\leq n+1$. If d=n, then X is covered by lines, whereas it seems hard to expect that X is covered by lines when d=n+1. However, if d=n+1, X is covered by conics. In this thesis, we study the existence problem of rational curves on a hypersurface including lines and conics. We also consider differential geometric property of quadric hypersurfaces.