On manifolds of small degree

Abstract

Let $X\subset P^n$ be a complex projective manifold of degree $d$ and arbitrary dimension. The main result of this paper gives a classification of such manifolds (assumed moreover to be connected, non-degenerate and linearly normal) in case $d\leq n$. As a by-product of the classification it follows that these manifolds are either rational or Fano. In particular, they are simply connected (hence regular) and of negative Kodaira dimension. Moreover, easy examples show that $d\leq n$ is the best possible bound for such properties to hold true. The proof of our theorem makes essential use of the adjunction mapping and, in particular, the main result of 'On varieties whose degree is small with respect to codimension' (Math.Ann. 271 1985) plays a crucial role in the argument.