  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 $0.79 \cdot n^{2/3}$.
