Geometric properties of projective manifolds of small degree

Abstract

The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb P^r$ of degree $d \leq r+2$, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalization of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb P^r$ of degree $d \leq r$ with counterexamples for $d=r+1, r+2$. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb P^r$ of dimension $n$ and degree $d \leq n(r-n)+2$ is Calabi-Yau, and give an example that shows this bound is also sharp.