Some remarks on smooth projective varieties of small degree and codimension

The purpose of this note is twofold. First, we give a quick proof of Ballico-Chiantini’s theorem stating that a Fano or Calabi-Yau variety of dimension at least 4 in codimension two is a complete intersection. Second, we improve Barth-Van de Ven’s result asserting that if the degree of a smooth projective variety of dimension $n$ is less than approximately $0.63\cdot n^{1\mathrm{/}2}$, then it is a complete intersection. We show that the degree bound can be improved to approximately $0.79\cdot n^{2\mathrm{/}3}$.