Hyperplane Sections of Hypersurfaces

Abstract

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset {\mathbb{P}}^{n+1}$ of degree $d$ over an algebraically closed field of characteristic zero, if $d>n>1$ and $(n,d)\ne (2,3),(3,4)$, then a general hyperplane section only admits finitely many others which are isomorphic to it.

18 pages, minor corrections. A case in the proof of Proposition 2.8 was overlooked (thanks to Dennis Tseng for pointing out this) and I withdraw the paper until that gap is filled