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 $0.63 \cdot n^{1/2}$, then it is a complete intersection. We show that the degree bound can be improved to approximately $0.79 \cdot n^{2/3}$.