On the hyperplane sections of a variety in projective space

Abstract

In this thesis, I am concerned with families of subvarieties of a fixed variety Y, and what bounds can be placed on the dimension of a family of isomorphic or birational subvarieties.

In particular I am most interested in families which arise as hyperplane sections, and I obtain a classification of smooth subvarieties of projective space, most of whose hyperplane sections are isomorphic. According to this classification any such variety must have very degenerate hyperplane sections.

I use two main tools, Lefschetz pencils and the scheme of Birational automorphisms.