Abstract 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 n^1/2, 0. 63 · n 1 / 2, then it is a complete intersection. We show that the degree bound can be improved to approximately 0. 79 n^2/3. 0. 79 · n 2 / 3.
Jinhyung Park (Thu,) studied this question.