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 2 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 · n1/2, 0.63 · n 1 / 2 , then it is a complete intersection. We show that the degree bound can be improved to approximately 0.79 · n2/3. 0.79 · n 2 / 3 .