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 P-r of degree d <= r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in P-r of degree d <= r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in P-r of dimension n and degree d <= n(r - n) + 2 is Calabi-Yau, and give an example that shows this bound is also sharp